Abstract
We study Choquard type equation of the form
where \(N\ge 3\), \(I_\alpha \) is the Riesz potential with \(\alpha \in (0,N)\), \(p>1\), \(q>2\) and \(\varepsilon \ge 0\). Equations of this type describe collective behaviour of self-interacting many-body systems. The nonlocal nonlinear term represents long-range attraction while the local nonlinear term represents short-range repulsion. In the first part of the paper for a nearly optimal range of parameters we prove the existence and study regularity and qualitative properties of positive groundstates of \((P_0)\) and of \((P_\varepsilon )\) with \(\varepsilon >0\). We also study the existence of a compactly supported groundstate for an integral Thomas–Fermi type equation associated to \((P_{\varepsilon })\). In the second part of the paper, for \(\varepsilon \rightarrow 0\) we identify six different asymptotic regimes and provide a characterisation of the limit profiles of the groundstates of \((P_\varepsilon )\) in each of the regimes. We also outline three different asymptotic regimes in the case \(\varepsilon \rightarrow \infty \). In one of the asymptotic regimes positive groundstates of \((P_\varepsilon )\) converge to a compactly supported Thomas–Fermi limit profile. This is a new and purely nonlocal phenomenon that can not be observed in the local prototype case of \((P_\varepsilon )\) with \(\alpha =0\). In particular, this provides a justification for the Thomas–Fermi approximation in astrophysical models of self-gravitating Bose–Einstein condensate.
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1 Introduction
1.1 Background
We are concerned with the asymptotic properties of positive groundstate solutions of the Choquard type equation
where \(N\ge 3\), \(p>1\), \(q>2\) and \({\varepsilon }\ge 0\). Here \(I_\alpha (x):=A_\alpha |x|^{-(N-\alpha )}\) is the Riesz potential with \(\alpha \in (0,N)\) and \(*\) denotes the standard convolution in \({\mathbb {R}}^N\). The choice of the normalisation constant \(A_\alpha :=\frac{\Gamma ((N-\alpha )/2)}{\pi ^{N/2}2^{\alpha }\Gamma (\alpha /2)}\) ensures that \(I_\alpha (x)\) could be interpreted as the Green function of \((-\Delta )^{\alpha /2}\) in \({\mathbb {R}}^N\), and that the semigroup property \(I_{\alpha +\beta }=I_\alpha *I_\beta \) holds for all \(\alpha ,\beta \in (0,N)\) such that \(\alpha +\beta <N\), see for example [22, pp. 73–74].
Equation
is often known as the Choquard equation and had been studied extensively during the last decade, see [42] for a survey. In this work we are interested in the case when the standard Choquard equation is modified by including the local repulsive \(|u|^{q-2}u\) term.
If \(u_{\varepsilon }\) is a solution of (\(P_{\varepsilon }\)) with \(N=3\), \(\alpha =2\), \(p=2\) and \(q=4\) then \(\psi (t,x):=e^{i{\varepsilon }t}u_{\varepsilon }(x)\) is a standing wave solution of the time-dependent equation
Equation of this form models, in particular, self-gravitating Bose–Einstein condensates with repulsive short-range interactions, which describe astrophysical objects such as boson stars and, presumably, dark matter galactic halos. In this context, (1.1) was introduced and studied under the name of Gross–Pitaevskii–Poisson equation in [8, 16, 53], see a survey paper [17].
More generally, Eq. (\(P_{\varepsilon }\)) can be seen as a stationary NLS with an attractive long range interaction, represented by the nonlocal term, coupled with a repulsive short range interaction, represented by the local nonlinearity. While for the most of the relevant physical applications \(p=2\), the values \(p\ne 2\) may appear in several relativistic models of the density functional theory [2,3,4].
In this work we are specifically interested in the case where \({\varepsilon }>0\) is a small (or large) parameter and all other parameters are fixed. Our main goal is to understand the behaviour of groundstate solutions of (\(P_{\varepsilon }\)) when \({\varepsilon }\rightarrow 0\). We also discuss the case \({\varepsilon }\rightarrow \infty \), which is to some extent dual to \({\varepsilon }\rightarrow 0\). The local prototype of (\(P_{\varepsilon }\)) and a formal limit of (\(P_{\varepsilon }\)) as \(\alpha \rightarrow 0\) is the equation
It is well-known that this equation admits a unique positive solution in \(H^1({\mathbb {R}}^N)\) for any \(1<2p<q<\infty \) provided that \({\varepsilon }>0\) is sufficiently small, and has no finite energy solutions for large \({\varepsilon }\). This result goes back to Strauss [51, Example 2] and Berestycki and Lions [6, Example 2] (see [41, Theorem A] for a precise existence statement and further references). A complete characterization of all possible asymptotic regimes in (1.2) as \({\varepsilon }\rightarrow 0\) was obtained in [41], see also earlier work [46]. Essentially, three different limit regimes were identified in [41], depending on whether p is less, equal or bigger than the critical Sobolev exponent \(p^*=\frac{N}{N-2}\). Recently, (1.2) had been revisited in [28] where nondegeneracy of ground-sates and sharp asymptotics of the \(L^2\)-norm of the ground states as \({\varepsilon }\rightarrow 0\) had been described in connection with the uniqueness conjecture in the \(L^2\)-constraint minimization problem associated to (1.2). See also [39], where the same problem is studied with the opposite sign of the \(|u|^{q-2}u\)-term.
1.2 Existence and properties of groundstates for (\(P_{\varepsilon }\))
We are not aware of a systematic study of ground-sates of Choquard Eq. (\(P_{\varepsilon }\)). First existence results seem to appear in [45] in the case \(N=3\), \(\alpha =2\), \(p=2\). See also [7, 23, 27, 29] and references therein for further results which however do not cover the optimal ranges of parameters. The planar case with the logarithmic convolution kernel was studied in [19] but since the kernel is sign-changing this requires different techniques. Near optimal existence results for the Choquard equation of type (\(P_{\varepsilon }\)) with an attractive local perturbation (the opposite sign of the local nonlinear term) were recently obtained in [30, 31].
Our first goal in this work is to establish the existence of ground-sate solutions of Choquard Eq. (\(P_{\varepsilon }\)) for an optimal range of parameters. By a groundstate solution of (\(P_{\varepsilon }\)) we understand a weak solution \(u \in H^1({\mathbb {R}}^N)\cap L^q({\mathbb {R}}^N)\) which has a minimal energy
amongst all nontrivial finite energy solutions of (\(P_{\varepsilon }\)). Remarkably, and in contrast with its local prototype (1.2), we prove that ground states for (\(P_{\varepsilon }\)) exist for every \({\varepsilon }>0\). We also establish some qualitative properties of the solutions (\(P_{\varepsilon }\)) such as regularity and decay at infinity. These properties are similar to the standard Choquard Eq. (\(\mathscr {C}\)). Note that we do not study the uniqueness or non-degeneracy of the groundstates of (\(P_{\varepsilon }\)) and we are not aware of any even partial results in this direction. We believe this is a very difficult open problem. Our results do not rely and do not require the uniqueness or non-degeneracy.
Essential tools to control the nonlocal term in \(\mathcal {I}_{\varepsilon }\) are the Hardy–Littlewood–Sobolev (HLS) inequality
which is valid for any \(p\ge 1\) (and \(\mathcal {C}_\alpha \) is independent of p); and the Sobolev inequality
where \(2^*=\frac{2N}{N-2}\) is the critical Sobolev exponent and \(D^1({\mathbb {R}}^N)\) denotes the homogeneous Sobolev space with the norm \(\Vert u\Vert _{D^1({\mathbb {R}}^N)}=\Vert \nabla u\Vert _{L^2}\). The values of the sharp constants \(\mathcal {S}_*>0\) and \(\mathcal {C}_{\alpha }>0\) are known explicitly [34, 35]. HLS and Sobolev inequalities can be used to control the nonlocal term in the two cases:
-
if \(\frac{N+\alpha }{N}\le p\le \frac{N+\alpha }{N-2}\) then \(L^\frac{2Np}{N+\alpha }({\mathbb {R}}^N)\subset L^2\cap L^{2^*}({\mathbb {R}}^N)\)
-
if \(p\ge \frac{N+\alpha }{N}\) and \(q\ge \frac{2Np}{N+\alpha }\) then \(L^\frac{2Np}{N+\alpha }({\mathbb {R}}^N)\subset L^2\cap L^q({\mathbb {R}}^N)\)
The two cases have non-empty intersection but this is not significant for us at this moment. In each of these two cases, \(\mathcal I_{\varepsilon }: H^1({\mathbb {R}}^N)\cap L^q({\mathbb {R}}^N)\rightarrow {\mathbb {R}}\) is well defined and critical points of \(\mathcal {I}_{{\varepsilon }}\) are solutions of (\(P_{\varepsilon }\)).
Our main existence result for (\(P_{\varepsilon }\)) is the following.
Theorem 1.1
Let \(\frac{N+\alpha }{N}<p<\frac{N+\alpha }{N-2}\) and \(q>2\), or \(p\ge \frac{N+\alpha }{N-2}\) and \(q>\frac{2Np}{N+\alpha }\). Then for each \({\varepsilon }>0\), Eq. (\(P_{\varepsilon }\)) admits a positive spherically symmetric ground state solution \(u_{\varepsilon }\in H^1\cap L^1\cap C^2({\mathbb {R}}^N)\) that is a monotone decreasing function of |x|. Moreover, there exists \(C_{\varepsilon }>0\) such that
-
if \(p > 2\),
$$\begin{aligned} \lim _{|x | \rightarrow \infty } u_{\varepsilon }(x) |x |^{\frac{N - 1}{2}} e^{\sqrt{{\varepsilon }}|x |}=C_{\varepsilon }, \end{aligned}$$ -
if \(p = 2\),
$$\begin{aligned} \lim _{|x | \rightarrow \infty } u_{\varepsilon }(x) |x |^{\frac{N - 1}{2}} \exp \int _{\nu }^{|x |} \sqrt{{\varepsilon }- \tfrac{\nu ^{N - \alpha }}{s^{N - \alpha }}} \,ds=C_{\varepsilon },\quad \text {where}\quad \nu :=\left( A_\alpha \Vert u_{\varepsilon }\Vert _2^2\right) ^\frac{1}{N - \alpha }, \end{aligned}$$ -
if \(p < 2\),
$$\begin{aligned} \lim _{x\rightarrow \infty }u_{\varepsilon }(x)|x|^\frac{N-\alpha }{2-p}=\left( {\varepsilon }^{-1}A_\alpha \Vert u_{\varepsilon }\Vert _p^p\right) ^{\frac{1}{2-p}}.\end{aligned}$$
The existence range of Theorem 1.1 is optimal. This follows from the Pohožaev identity argument, see Corollary 4.1. We emphasise that no upper restrictions on p and q are needed and in particular, q could take Sobolev supercritical values, i.e. \(q>2^*\) (see Fig. 1). The decay rates of ground states at infinity are exactly the same as in the standard Choquard case, compare Theorem 2.4 below or [43, Theorem 4]. For a discussion of the implicit exponential decay in the case \(p=2\) we refer to [43, pp. 157–158] or [44, Section 6.1].
Our main goal in this paper is to understand and classify the asymptotic profiles as \({\varepsilon }\rightarrow 0\) and \({\varepsilon }\rightarrow \infty \) of the groundstates \(u_{\varepsilon }\), constructed in Theorem 1.1. Remarkably, our study uncovers a novel and rather complicated limit structure of the problem, with six different limit equations (see Fig. 1) as \({\varepsilon }\rightarrow 0\):
-
Formal limit when the family of ground states \(u_{\varepsilon }\) converges to a groundstate of the formal limit equation
The existence and qualitative properties of groundstate for (\(P_0\)) for the optimal range of parameters is new and is studied in Sect. 5, see Theorem 2.1. The convergence of the groundstates to the limit profile is proved in Theorem 2.2.
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Choquard limit when the rescaled family
$$\begin{aligned} v_{\varepsilon }(x):={\varepsilon }^{-\frac{2+\alpha }{4(p-1)}}u_{\varepsilon }\left( {\varepsilon }^{-\frac{1}{2}}x\right) \end{aligned}$$converges to a groundstate of the standard Choquard equation
which was studied in [43]. The convergence is proved in Theorem 2.5.
-
Thomas–Fermi limit when the rescaled family
$$\begin{aligned}v_{\varepsilon }(x):={\varepsilon }^{-\frac{1}{q-2}}u_{\varepsilon }\left( {\varepsilon }^{-\frac{4-q}{\alpha (q-2)}} x\right) \end{aligned}$$converges to a groundstate of the Thomas–Fermi type integral equation
The existence and qualitative properties of groundstate for (\(T\!F\)) for \(p\ne 2\) will be studied in the forthcoming work [26]. In this paper we consider only the case \(p=2\) which is well known in the literature when \(\alpha =2\) [1, 5, 36] and was studied recently in [11, 12] for the general \(\alpha \in (0,N)\), yet for the range of powers q which is incompatible with our assumptions. In Theorem 2.6 we prove the existence and some qualitative properties of a groundstate for (\(T\!F\)) with \(p=2\) for the optimal range \(q>\frac{4N}{N+\alpha }\). This extends some of the existence results in [11, 12]. The convergence of \(v_{\varepsilon }\) to a groundstate of (\(T\!F\)) is proved in Theorem 2.7 for \(p=2\) and \(\alpha =2\) (the general case \(p\ne 2\) and \(\alpha \ne 2\) will be studied in [26]). Remarkably, for \(p=2\) the limit groundstates of (\(T\!F\)) are compactly supported, so the rescaled groundstates \(v_{\varepsilon }\) develops a steep “corner layer” as \({\varepsilon }\rightarrow 0\)!
-
Critical Choquard regime, when the family of ground states \(u_{\varepsilon }\) converges after an implicit rescaling
$$\begin{aligned} v_{\varepsilon }(x):=\lambda _{\varepsilon }^\frac{N-2}{2}u_{\varepsilon }(\lambda _{\varepsilon }x)\end{aligned}$$to a groundstate of the critical Choquard equation
A detailed characterisation of the ground states of (\(\mathscr {C}_{HL}\)) was recently obtained in [18, 21]. In Theorem 2.8 we derive a sharp two-sided asymptotic characterisation of the rescaling \(\lambda _{\varepsilon }\), following the ideas developed in the local case in [41].
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Self-similar regime \(q=2\frac{2p+\alpha }{2+\alpha }\), when ground states \(u_{\varepsilon }\) are obtained as rescalings of the groundstate \(u_1\), i.e.
$$\begin{aligned} u_1(x)={\varepsilon }^{-\frac{2+\alpha }{4(p-1)}}u_{\varepsilon }\left( {\varepsilon }^{-\frac{1}{2}}x\right) \end{aligned}$$ -
the Critical Thomas–Fermi regime, when the family of ground states \(u_{\varepsilon }\) converges after an implicit rescaling
$$\begin{aligned}v_{\varepsilon }(x):=\lambda _{\varepsilon }^\frac{N+\alpha }{2p}u_{\varepsilon }(\lambda _{\varepsilon }x)\end{aligned}$$to a groundstate of the critical Thomas–Fermi equation
Groundstates of this equation correspond to the minimizers of the Hardy–Littlewood–Sobolev inequality and completely characterised by Lieb in [34]. In Theorem 2.9 we derive a two-sided asymptotic characterisation of the rescaling \(\lambda _{\varepsilon }\).
Self-similar, Thomas–Fermi and critical Thomas–Fermi regimes are specific to the nonlocal case only. When \(\alpha =0\) they all “collapse” into the case \(p=q\), which is degenerate for the local prototype Eq. (1.2). Three other regimes could be traced back to the local Eq. (1.2) studied in [41].
When \({\varepsilon }\rightarrow \infty \) the limit structure is simpler. Only Choquard, Thomas–Fermi and self-similar regimes are relevant (see Fig. 2) and there are no critical regimes. In particular, the Thomas–Fermi limit with \({\varepsilon }\rightarrow \infty \) appears in the study of the stationary Gross–Pitaevskii–Poisson Eq. (1.1), see Remark 3.1.
The precise statements of our results for \({\varepsilon }\rightarrow 0\) are given in Sect. 2. In Sect. 3 we outline the results for \({\varepsilon }\rightarrow \infty \) and discuss the connection with astrophysical models of self-gravitating Bose–Einstein condensate. In Sect. 4 we prove Theorem 1.1. In Sects. 5 and 6 we establish the existence and basic properties of groundstates for the “zero-mass” limit equation \((P_0)\) and for the Thomas–Fermi Eq. (\(T\!F\)). In Sects. 7 and 8 we study the asymptotic profiles of the groundstates of (\(P_{\varepsilon }\)) in the non-critical and critical regimes respectively. Finally, in the Appendix we discuss a contraction inequality which was communicated to us by Augusto Ponce and which we used as a key tool in several regularity proofs.
1.2.1 Asymptotic notations
For real valued functions \(f(t), g(t) \ge 0\) defined on a subset of \({\mathbb {R}}_+\), we write:
-
\(f(t)\lesssim g(t)\) if there exists \(C>0\) independent of t such that \(f(t) \le C g(t)\);
-
\(f(t) > rsim g(t)\) if \(g(t)\lesssim f(t)\);
-
\(f(t)\sim g(t)\) if \(f(t)\lesssim g(t)\) and \(f(t) > rsim g(t)\);
-
\(f(t)\simeq g(t)\) if \(f(t) \sim g(t)\) and \(\lim _{t\rightarrow 0}\frac{f(t)}{g(t)}=1\).
Bearing in mind that \(f(t),g(t) \ge 0\), we write \(f(t)=\mathcal {O}(g(t))\) if \(f(t)\sim g(t)\), and \(f(t)=o(g(t))\) if \(\lim \frac{f(t)}{g(t)}=0\). As usual, \(B_R=\{x\in {\mathbb {R}}^N:|x|<R\}\) and \(C,c,c_1\) etc., denote generic positive constants.
2 Asymptotic profiles as \({\varepsilon }\rightarrow 0\)
Our main goal in this work is to understand the asymptotic behaviour of the constructed in Theorem 1.1 groundstate solutions \(u_{\varepsilon }\) of (\(P_{\varepsilon }\)) in the limits \({\varepsilon }\rightarrow 0\) and \({\varepsilon }\rightarrow \infty \).
2.1 Formal limit \((P_0)\)
Loosely speaking, the elliptic regularity implies that \(u_{\varepsilon }\) converges as \({\varepsilon }\rightarrow 0\) to a nonnegative radial solution of the formal limit equation
However, (\(P_0\)) becomes a meaningful limit equation for (\(P_{\varepsilon }\)) only in the situation when (\(P_0\)) admits a nontrivial nonnegative solution. Otherwise the information that \(u_{\varepsilon }\) converges to zero (trivial solution of (\(P_0\))) does not reveal any information about the limit profile of \(u_{\varepsilon }\). We prove in this work the following existence result for (\(P_0\)).
Theorem 2.1
Let \(\frac{N+\alpha }{N}<p<\frac{N+\alpha }{N-2}\) and \(2<q<\frac{2Np}{N+\alpha }\), or \(p>\frac{N+\alpha }{N-2}\) and \(q>\frac{2Np}{N+\alpha }\). Then Eq. (\(P_0\)) admits a positive spherically symmetric groundstate solution \(u_0 \in D^1\cap L^q\cap C^2({\mathbb {R}}^N)\) which is a monotone decreasing function of |x|. Moreover,
-
if \(p < \frac{N+\alpha }{N-2}\) then \(u\in L^1({\mathbb {R}}^N)\),
-
if \(p > \frac{N+\alpha }{N-2}\) then
$$\begin{aligned} u_0 > rsim |x|^{-(N-2)}\quad \text {as}\quad |x|\rightarrow \infty , \end{aligned}$$and if \(p>\max \left\{ \frac{N+\alpha }{N-2},\frac{2}{3}\left( 1+\frac{N+\alpha }{N-2}\right) \right\} \) then
$$\begin{aligned} u_0\sim |x|^{-(N-2)}\quad \text {as}\quad |x|\rightarrow \infty . \end{aligned}$$
The restrictions on p and q in the existence part of the theorem ensures that the energy \(\mathcal {I}_0\) which corresponds to (\(P_0\)) is well-defined on \(D^1({\mathbb {R}}^N)\cap L^q({\mathbb {R}}^N)\), see (5.1) below. A Pohožaev identity argument (see Remark 5.2) confirms that the existence range in Theorem 2.1 is optimal, with the exception of the “double critical” point \(p=\frac{N+\alpha }{N-2}\) and \(q=\frac{2Np}{N+\alpha }=2^*\) on the (p, q)-plane (see Remark 5.3).
Note that
The upper bound on \(u_0\) in the case \(\frac{N+\alpha }{N-2}<2\) and \(\frac{N+\alpha }{N-2}<p\le \frac{2}{3}\left( 1+\frac{N+\alpha }{N-2}\right) \) remains open. We conjecture that our restriction on p for the upper decay estimate is technical and that \(u_0\sim |x|^{-(N-2)}\) as \(|x|\rightarrow \infty \) for all \(p>\frac{N+\alpha }{N-2}\).
Observe that the energy \(\mathcal {I}_0\) is well posed in the space \(D^1({\mathbb {R}}^N)\cap L^q({\mathbb {R}}^N)\), while \(\mathcal {I}_{\varepsilon }\) with \({\varepsilon }>0\) is well-posed on \(H^1({\mathbb {R}}^N)\cap L^q({\mathbb {R}}^N)\). Since \(H^1({\mathbb {R}}^N)\subsetneq D^1({\mathbb {R}}^N)\), small perturbation arguments in the spirit of the Lyapunov–Schmidt reduction are not directly applicable to the family \(\mathcal {I}_{\varepsilon }\) in the limit \({\varepsilon }\rightarrow 0\). Using direct variational analysis based on the comparison of the groundstate energy levels for two problems, we establish the following result.
Theorem 2.2
(Formal limit (\(P_0\))) Let \(\frac{N+\alpha }{N}<p<\frac{N+\alpha }{N-2}\) and \(2<q<\frac{2Np}{N+\alpha }\), or \(p>\max \left\{ \frac{N+\alpha }{N-2},\frac{2}{3}\left( 1+\frac{N+\alpha }{N-2}\right) \right\} \) and \(q>\frac{2Np}{N+\alpha }\).
Then as \({\varepsilon }\rightarrow 0\), the family of ground states \(u_{\varepsilon }\) of \((P_{\varepsilon })\) converges in \(D^1({\mathbb {R}}^N)\) and \(L^q({\mathbb {R}}^N)\) to a positive spherically symmetric ground state solution \(u_0\in D^1\cap L^q({\mathbb {R}}^N)\) of the formal limit Eq. (\(P_0\)). Moreover, \({\varepsilon }\Vert u_{\varepsilon }\Vert _2^2\rightarrow 0\).
The restriction \(p>\max \left\{ \frac{N+\alpha }{N-2},\frac{2}{3}\left( 1+\frac{N+\alpha }{N-2}\right) \right\} \) is related only to the upper decay bound on \(u_0\) in Theorem 2.1, i.e. we could establish the convergence of \(u_{\varepsilon }\) to \(u_0\) for \(p>\frac{N+\alpha }{N-2}\) as soon as we know that \(u_0\sim |x|^{-(N-2)}\) as \(|x|\rightarrow \infty \).
2.2 Rescaled limits
When \(\frac{N+\alpha }{N}<p\le \frac{N+\alpha }{N-2}\) and \(q\ge \frac{2Np}{N+\alpha }\) the formal limit problem (\(P_0\)) has no nontrivial sufficiently regular finite energy solutions (see Remark 5.3). As a consequence, \(u_{\varepsilon }\) converges uniformly on compact sets to zero. We are going to show that in these regimes \(u_{\varepsilon }\) converges to a positive solution of a limit equation after a rescaling
for specific choices of \(s,t\in {\mathbb {R}}\). The rescaling transforms (\(P_{\varepsilon }\)) into the equation
If \(q\ne 2\frac{2p+\alpha }{2+\alpha }\) then there are three natural possibilities to choose s and t, each achieving the balance of three different terms in (2.1). Note that the choice \({\varepsilon }^{-s-2t}={\varepsilon }^{-(2p-1)s+\alpha t}={\varepsilon }^{-(q-1)s}\) leads to \(s=t=0\), when (2.1) reduces to the original equation \((P_{\varepsilon })\).
I. First rescaling. The choice \({\varepsilon }^{-s-2t}={\varepsilon }^{1-s}={\varepsilon }^{-(2p-1)s+\alpha t}\) leads to \(s=-\frac{2+\alpha }{4(p-1)}\), \(t=-1/2\) and rescaled equation
If \(q>2\frac{2p+\alpha }{2+\alpha }\), we have \(\lim _{{\varepsilon }\rightarrow 0}{\varepsilon }^{\frac{q(2+\alpha )-2(2p+\alpha )}{4(p-1)}}=0\), and we obtain as a formal limit the Choquard equation
II. Second rescaling. The choice \({\varepsilon }^{1-s}={\varepsilon }^{-(2p-1)s+\alpha t}={\varepsilon }^{-(q-1)s}\) leads to \(s=-\frac{1}{q-2}\), \(t=-\frac{2p-q}{\alpha (q-2)}\) and rescaled equation
If \(2<q<2\frac{2p+\alpha }{2+\alpha }\) we have \(\lim _{{\varepsilon }\rightarrow 0}{\varepsilon }^{\frac{2(2p+\alpha )-q(2+\alpha )}{\alpha (q-2)}}=0\) and we obtain as a formal limit the Thomas–Fermi type integral equation
III. Third rescaling The choice \({\varepsilon }^{-s-2t}={\varepsilon }^{1-s}={\varepsilon }^{-(q-1)s}\) leads to \(s=-\frac{1}{q-2}\), \(t=-\frac{1}{2}\) and rescaled equation
If \(2<q<2\frac{2p+\alpha }{2+\alpha }\) we have \(\lim _{{\varepsilon }\rightarrow 0}{\varepsilon }^{\frac{2(2p+\alpha )-q(\alpha +2)}{2(q-2)}}=0\), and we obtain as a formal limit the nonlinear local equation
Such equation has no nonzero finite energy solutions and we rule out the third rescaling as trivial.
2.3 Self-similar regime \(q=2\frac{2p+\alpha }{2+\alpha }\)
In this special case the choice \(s=-\frac{2+\alpha }{4(p-1)}\) and \(t=-1/2\) leads to the balance of all four terms in (2.1), i.e. \({\varepsilon }^{-s-2t}={\varepsilon }^{1-s}={\varepsilon }^{-(2p-1)s+\alpha t}={\varepsilon }^{-(q-1)s}\). The rescaled equation in this case becomes
and any solution of \((P_{\varepsilon })\) is a rescaling of a solution of \((P_1)\). For completeness, we include this obvious observation in the following statement.
Theorem 2.3
(Self-similar regime) Let \(\frac{N+\alpha }{N}<p<\frac{N+\alpha }{N-2}\) and \(q=2\frac{2p+\alpha }{2+\alpha }\). Then for any \({\varepsilon }>0\),
where \(u_1\) is a ground state solution of \((P_1)\).
2.4 First rescaling: Choquard limit
The following result describes the existence region and some qualitative properties of the groundstates of (\(\mathscr {C}\)).
Theorem 2.4
[43, Theorem 4] Let \(\frac{N+\alpha }{N}<p<\frac{N+\alpha }{N-2}\). Then Choquard’s Eq. (\(\mathscr {C}\)) admits a positive spherically symmetric ground state solution \(v\in H^1\cap L^1\cap C^2({\mathbb {R}}^N)\) such that v(x) is a monotone decreasing function of |x|. Moreover, there exists \(C>0\) such that
-
if \(p > 2\),
$$\begin{aligned} \lim _{|x | \rightarrow \infty } v(x) |x |^{\frac{N - 1}{2}} e^{|x |}=C, \end{aligned}$$ -
if \(p = 2\),
$$\begin{aligned} \lim _{|x | \rightarrow \infty } v(x) |x |^{\frac{N - 1}{2}} \exp \int _{\nu }^{|x |} \sqrt{1 - \tfrac{\nu ^{N - \alpha }}{s^{N - \alpha }}} \,ds=C,\quad \text {where}\quad \nu :=\left( A_\alpha \Vert v\Vert _2^2\right) ^\frac{1}{N - \alpha } , \end{aligned}$$ -
if \(p < 2\),
$$\begin{aligned} \lim _{|x|\rightarrow \infty }v(x)|x|^\frac{N-\alpha }{2-p}=\left( A_\alpha \Vert v\Vert _p^p\right) ^{\frac{1}{2-p}}.\end{aligned}$$
The existence interval in this theorem is sharp, in the sense that (\(\mathscr {C}\)) does not have finite energy solutions for \(p\not \in \left( \frac{N+\alpha }{N},\frac{N+\alpha }{N-2}\right) \). The uniqueness of the ground state solution is known only for \(N=3\), \(p=2\) and \(\alpha =2\) [33] and several other special cases [38, 49, 54].
In this paper we prove that after the 1st rescaling, groundstates of (\(P_{\varepsilon }\)) converge to the groundstates of the Choquard Eq. (\(\mathscr {C}\)), as soon as (\(\mathscr {C}\)) admits a nontrivial groundstate.
Theorem 2.5
(Choquard limit) Let \(\frac{N+\alpha }{N}<p<\frac{N+\alpha }{N-2}\) and \(q>2\frac{2p+\alpha }{2+\alpha }\). As \({\varepsilon }\rightarrow 0\), the rescaled family of ground states
converges in \(D^1({\mathbb {R}}^N)\) and \(L^q({\mathbb {R}}^N)\) to a positive spherically symmetric ground state solution \(v_0\in D^1\cap L^q({\mathbb {R}}^N)\) of the Choquard Eq. (\(\mathscr {C}\)).
2.5 2nd rescaling: Thomas–Fermi limit for \(p=2\)
In this paper we consider the 2nd rescaling regime only in the case \(p=2\). The general case \(p\ne 2\) is studied in the forthcoming work [26].
When \(p=2\) the formal limit equation for (\(P_{\varepsilon }\)) in the 2nd rescaling is the Thomas–Fermi type integral equation
One of the possible ways to write the variational problem that leads to (\(T\!F\)) after a rescaling is
By a groundstate of (\(T\!F\)) we understand a rescaling of a nonnegative minimizer for \(s_{T\!F}\) that satisfies the limit Eq. (\(T\!F\)).
To study nonnegative minimizers of the \(s_{T\!F}\) it is convenient to substitute
for an equivalent representation
For \(m>m_c:=2-\alpha /N\) it is not difficult to see that, after a rescaling, minimizers for \(s_{T\!F}\) are in the one-to-one correspondence with the minimizers of
The existence and qualitative properties of minimizers for \(\sigma _{T\!F}\) in the case \(N=3\), \(\alpha =2\) and for \(m>4/3\) is classical and goes back to [1, 36]. The case \(N\ge 2\), \(\alpha \in (0,N)\) and \(m>m_c\) it is a recent study by Carrillo et al. [12]. If \(m<m_c\) then \(\sigma _{T\!F}=-\infty \) by scaling, while \(m=m_c\) is the \(L^1\)-critical exponent for \(\sigma _{T\!F}\) (this case is studied in [11]). Note that \(m_c>\frac{4+\alpha }{2+\alpha }\) so in the 2nd rescaling regime we always have \(\sigma _{T\!F}=-\infty \) when \({\varepsilon }\rightarrow 0\)!
In the next theorem we show that, unlike for \(\sigma _{T\!F}\), minimization for \(s_{T\!F}\) is possible for all \(m>\frac{2N}{N+\alpha }\). The existence and qualitative properties of the minimizers are summarised below.
Theorem 2.6
(Thomas–Fermi groundstate) Let \(m>\frac{2N}{N+\alpha }\). Then \(s_{T\!F}>0\) and there exists a nonnegative spherically symmetric nonincreasing minimizer \(\rho _*\in L^1\cap L^\infty ({\mathbb {R}}^N)\) for \(s_{T\!F}\). The minimizer \(\rho _*\) satisfies the virial identity
and the Thomas–Fermi equation
Moreover, \(\mathrm {supp}(\rho _*)=\bar{B}_{R_*}\) for some \(R_*\in (0,+\infty )\), \(\rho _*\) is \(C^\infty \) inside the support, and \(\rho _*\in C^{0,\gamma }({\mathbb {R}}^N)\) with \(\gamma =\min \{1,\frac{1}{m-1}\}\) if \(\alpha >\left( \frac{m-2}{m-1}\right) _+\), or for any \(\gamma <\frac{\alpha }{m-2}\) if \(m>2\) and \(\alpha \le \left( \frac{m-2}{m-1}\right) _+\). Moreover, if \(\alpha >\left( \frac{m-2}{m-1}\right) _+\) then \(I_\alpha *\rho _*\in C^{0,1}({\mathbb {R}}^N)\) and \(\rho _*^{m-1}\in C^{0,1}({\mathbb {R}}^N)\). Meanwhile,
is a nonnegative spherically symmetric nonincreasing ground state solution of the Thomas–Fermi Eq. (\(T\!F\)).
Only the existence part of the theorem requires a proof. The Euler–Lagrange equation, regularity and qualitative properties of the minimizers could be obtained by adaptations of the arguments developed for \(m>m_c\) in [11, 12]. We outline the arguments in Sect. 7.3.
In the case \(m\ge m_c\) the uniqueness of the minimizer for \(\sigma _{T\!F}\) was recently proved in [10] for \(\alpha <2\), see also [13] for \(\alpha =2\) and a survey of earlier results in this direction. For the full range \(m>\frac{2N}{N+\alpha }\) and for \(\alpha <2\) the uniqueness of a bounded radially nonincreasing solution for the Euler–Lagrange Eq. (2.5) (and hence the uniqueness of the minimizer \(\rho _*\) for \(s_{T\!F}\)) is the recent result in [14, Theorem 1.1 and Proposition 5.4]. For \(\alpha =2\) the same follows from [24, Lemma 5]. For \(\alpha \in (2,N)\) the uniqueness of the minimizer for \(s_{T\!F}\) or for \(\sigma _{T\!F}\) seems to be open at present.
Next we prove that in the special case \(\alpha =2\), groundstates of (\(P_{\varepsilon }\)) converge to a groundstate of the Thomas–Fermi Eq. (\(T\!F\)).
Theorem 2.7
(Thomas–Fermi limit for \(\alpha =2\)) Let \(N\le 5\), \(p=2\), \(\alpha =2\) and \(\frac{4N}{N+2}<q<3\). As \({\varepsilon }\rightarrow 0\), the rescaled family of ground states
converges in \(L^2({\mathbb {R}}^N)\) and \(L^q({\mathbb {R}}^N)\) to a nonnegative spherically symmetric compactly supported ground state solution \(v_0\in L^2\cap L^q({\mathbb {R}}^N)\) of the Thomas–Fermi Eq. (\(T\!F\)).
Remark 2.1
While the uniqueness of the Thomas–Fermi groundstate \(v_0\) for \(\alpha >2\) is generally open, it is clear from the proof of Theorem 2.6 that every ground state of (\(T\!F\)) must have the same regularity and compact support properties as stated in Theorem 2.6. In particular, \(v_{\varepsilon }\) always exhibits as \({\varepsilon }\rightarrow 0\) a “corner layer” near the boundary of the support of the limit groundstate of (\(T\!F\)).
Remark 2.2
The case \(p\ne 2\) and \(\alpha \ne 2\) is studied in our forthcoming paper [26]. We are going to show that the minimization problem
admits a nonnegative spherically symmetric nonincreasing minimizer \(v_*\in L^1\cap L^\infty ({\mathbb {R}}^N)\) for any \(p>\frac{N+\alpha }{N}\) and \(q>\frac{2Np}{N+\alpha }\) and this range is optimal. Moreover,
-
(a)
if \(p\ge 2\) then \(\mathrm {Supp}(\overline{v}_*)=B_{R_*}\) and \(\overline{v}_*=\lambda \chi _{B_{R_*}}+\phi \), where \(R_*>0\), \(\lambda >0\) if \(p>2\) or \(\lambda =0\) if \(p=2\), and \(\phi :B_R\rightarrow {\mathbb {R}}\) is Hölder continuous radially nonincreasing, \(\phi (0)>0\) and \(\lim _{|x|\rightarrow R_*}\phi (|x|)=0\);
-
(b)
if \(p<2\) then \(\overline{v}_*\in D^1({\mathbb {R}}^N)\) and \(\mathrm {Supp}(\overline{v}_*)={\mathbb {R}}^N\).
We will prove in [26] that such a minimizer is the limit of the rescaled groundstates \(v_{\varepsilon }(x)\) as \({\varepsilon }\rightarrow 0\) in the Thomas–Fermi regime.
2.6 Critical Choquard regime \(p=\frac{N+\alpha }{N-2}\)
When \(p=\frac{N+\alpha }{N-2}\) and \(q>\frac{2Np}{N+\alpha }=2^*\), neither (\(\mathscr {C}\)) nor \((P_0)\) have nontrivial solutions. We prove that in this case the limit equation for (\(P_{\varepsilon }\)) is given by the critical Choquard equation
A variational problem that leads to (\(\mathscr {C}_{HL}\)) can be written as
It is known [21, Lemma 1.1] that
where \(\mathcal {S}_*\) is the Sobolev constant in (1.4) and \(\mathcal {C}_\alpha \) is the Hardy–Littlewood–Sobolv constant in (1.3) (with \(p=\frac{N+\alpha }{N-2}\)).
By a groundstate of (\(\mathscr {C}_{HL}\)) we understand a rescaling of a positive minimizer for \(\mathcal {S}_{HL}\) that satisfies Eq. (\(\mathscr {C}_{HL}\)). Denote
a groundstate of the Emden–Fowler equation \(-\Delta U_*=U_*^{2^*-1}\) in \({\mathbb {R}}^N\). Then (see e.g. [21, Lemma 1.1]) all radial groundstates of (\(\mathscr {C}_{HL}\)) are given by the function
and the family of its rescalings
In fact, if \(N=3,4\) or if \(N\ge 5\) and \(\alpha \ge N-4\) then all finite energy solutions of (\(\mathscr {C}_{HL}\)) are given by the rescalings and translations of \(U_*\), see [21, Theorem 1.1].
We prove that in the critical Choquard regime the family of ground states \(u_{\varepsilon }\) converge in a suitable sense to V after an implicit rescaling \(\lambda _{\varepsilon }\). Note that \(V\in L^2({\mathbb {R}}^N)\) only if \(N\ge 5\) and hence the lower dimensions should be handled differently, as the \(L^2\)-norm of \(u_{\varepsilon }\) must blow up when \(N=3,4\). Our principal result is a sharp two-sided asymptotic estimate on the rescaling \(\lambda _{\varepsilon }\) as \({\varepsilon }\rightarrow 0\). Similar result in the local case \(\alpha =0\) was first observed in [46] and then rigorously established in [41]
Theorem 2.8
(Critical Choquard limit) Let \(p=\frac{N+\alpha }{N-2}\) and \(q>\frac{2Np}{N+2}=2^*\). There exists a rescaling \(\lambda _{\varepsilon }:(0,\infty )\rightarrow (0,\infty )\) such that as \({\varepsilon }\rightarrow 0\), the rescaled family of ground states
converges to V in \(D^1({\mathbb {R}}^N)\). Moreover, as \({\varepsilon }\rightarrow 0\),
2.7 Critical Thomas–Fermi regime \(q=\frac{2Np}{N+\alpha }\)
When \(\frac{N+\alpha }{N}<p<\frac{N+\alpha }{N-2}\) and \(q=\frac{2Np}{N+\alpha }\), neither (\(T\!F\)) nor \((P_0)\) have nontrivial solutions. We show that in this case the limit equation for (\(P_{\varepsilon }\)) is given by the critical Thomas–Fermi type equation
By a groundstate of (\(T\!F_*\)) we understand a positive solution of (\(T\!F_*\)) which is a rescaling of a nonnegative minimizer for the Hardy–Littlewood–Sobolev minimization problem
where \(\mathcal C_{\alpha }\) is the optimal constant in (1.3). It is known [34, Theorem 4.3] that all radial groundstates of (\(T\!F_*\)) are given by
for a constant \(\sigma _{\alpha ,N}>0\), and the family of rescalings
We prove that, similarly to the critical Choquard regime, in the critical Thomas–Fermi regime the family of ground states \(u_{\varepsilon }\) converge in a suitable sense to \(\widetilde{U}\) after an implicit rescaling \(\lambda _{\varepsilon }\). Note that \(\widetilde{U}\in L^2({\mathbb {R}}^N)\) and \(\widetilde{U}\in D^{1}({\mathbb {R}}^N)\) only if \(N\ge 4\), so \(N=3\) is now the only special dimension. Our main result in the critical Thomas–Fermi regime is the following.
Theorem 2.9
(Critical Thomas–Fermi limit) Let \(\frac{N+\alpha }{N}<p<\frac{N+\alpha }{N-2}\) and \(q=\frac{2Np}{N+\alpha }\). There exists a rescaling \(\lambda _{\varepsilon }:(0,\infty )\rightarrow (0,\infty )\) such that as \({\varepsilon }\rightarrow 0\), the rescaled family of ground states
converges to \(\widetilde{V}\) in \(L^q({\mathbb {R}}^N)\), where \(\widetilde{V}\) is defined by (8.17). Moreover, if \(N\ge 4\) then as \({\varepsilon }\rightarrow 0\),
while if \(N=3\) then
Remark 2.3
We expect that the upper asymptotic bounds (2.17) with \(p\ge \tfrac{2(3+\alpha )}{3}\) could be refined to match the lower bounds, but this remains open at the moment.
3 Asymptotic profiles as \({\varepsilon }\rightarrow \infty \) and Gross–Pitaevskii–Poisson model
The behaviour of ground states \(u_{\varepsilon }\) as \({\varepsilon }\rightarrow \infty \) is less complex than in the case \({\varepsilon }\rightarrow 0\). Only the 1st and the 2nd rescalings are meaningful, separated by the \(q=2\frac{2p+\alpha }{2+\alpha }\) line, however the limit equations “switch” compared to the case \({\varepsilon }\rightarrow 0\). There are no critical regimes.
Theorem 3.1
(Choquard limit) Let \(\frac{N+\alpha }{N}<p<\frac{N+\alpha }{N-2}\) and \(2<q<2\frac{2p+\alpha }{2+\alpha }\). As \({\varepsilon }\rightarrow \infty \), the rescaled family of ground states
converges in \(D^1({\mathbb {R}}^N)\) and \(L^q({\mathbb {R}}^N)\) to a positive spherically symmetric ground state solution \(v_0\in D^1\cap L^q({\mathbb {R}}^N)\) of the Choquard Eq. (\(\mathscr {C}\)).
Clearly, for \(q=\frac{2p+\alpha }{2+\alpha }\) the self-similar regime of Theorem 2.3 is valid also as \({\varepsilon }\rightarrow \infty \). For \(p=2\) and \(q>\frac{2p+\alpha }{2+\alpha }\) we have the following.
Theorem 3.2
(Thomas–Fermi limit for \(\alpha =2\)) Assume that \(p=2\) and \(\alpha =2\). Let \(N\le 5\) and \(q>3\), or \(N\ge 6\) and \(q>\frac{4N}{N+2}\). As \({\varepsilon }\rightarrow \infty \), the rescaled family of ground states
converges in \(L^2({\mathbb {R}}^N)\) and \(L^q({\mathbb {R}}^N)\) to a nonnegative spherically symmetric compactly supported ground state solution \(v_0\in L^2\cap L^q({\mathbb {R}}^N)\) of the Thomas–Fermi Eq. (\(T\!F\)).
The proofs of Theorems 3.2 and 3.1 are very similar to the proofs of Theorems 2.7 and 2.5. We only note that the proof on Theorem 3.2 will involve the estimate (7.25) with \(q\ge 4\) when the the right hand side of (7.25) blows-up. However the rate of the blow-up in (7.25) isn’t strong enough and all quantities involved in the proof remain under control. We leave the details to the interested readers.
Remark 3.1
Note that the nature of rescaling (3.2) changes when \(q=4\): for \(q>4\) the mass of \(u_{\varepsilon }\) concentrates near the origin, while for \(q<4\) it “escapes” to infinity. In particular, the stationary version of the Gross–Pitaevskii–Poisson Eq. (1.1) (\(q=4\), \(N=3\), \(\alpha =2\)) fits into the Thomas–Fermi regime as \({\varepsilon }\rightarrow \infty \). The rescaling (3.2) in this case takes the simple form \(v_{\varepsilon }(x)={\varepsilon }^{-1/2}u_{\varepsilon }(x)\) and \(u_{\varepsilon }(x)\approx \sqrt{{\varepsilon }}v_0(x)\), or we can say that \(u_{\varepsilon }\) concentrates towards the compactly supported \(v_0\). This is precisely the phenomenon which was already observed in [8, 53], where the radius of the support of \(v_0\) has the meaning of the radius of self-gravitating Bose–Einstein condensate, see [16]. The limit minimization problem \(s_{T\!F}\) in the Gross–Pitaevskii–Poisson Eq. (1.1) case becomes
and the Euler–Lagrange Eq. (2.5) in this case is linear inside the support of \(\rho \):
To find explicitly the solution of (3.3) constructed in Theorem 2.6, we use the \(\frac{\sin (|x|)}{|x|}\) ansatz as in [8, 53, 15, p. 92].
For \(\lambda >0\) and \(|x|\le \pi /\lambda \) consider the family
Then \(-\Delta \rho _\lambda =\lambda ^2\rho _\lambda \) in \(B_{\pi /\lambda }\), and
We compute
Optimising in \(\lambda >0\), we find that
and the minimum occurs at the optimal \(\lambda _*=\left( \frac{\pi ^2}{2}\right) ^{1/5}\). Taking into account the uniqueness of the spherically symmetric nonincreasing minimizer for \(s_{T\!F}\) in the case \(\alpha =2\), which follows from [13, Theorem 1.2] (see also [14, Lemma 5.2]), the function
is the unique spherically symmetric nonincreasing minimizer for \(s_{T\!F}\) and a solution of (3.3).
The solution of the limit Thomas–Fermi Eq. (\(T\!F\)), which is written in this case as
is given by the rescaled function in (2.6),
This is (up to the physical constants) the Thomas–Fermi approximation solution for self-gravitating BEC observed in [8, 16, 53] and the support radius \(R_0=\pi \) is the approximate radius of the BEC star. Our Theorem 3.2 provides a rigorous justification for the convergence of the Thomas–Fermi approximation.
4 Existence and properties of groundstates for (\(P_{\varepsilon }\))
4.1 Variational setup
It is a standard consequence of Sobolev and Hardy–Littlewood–Sobolev (HLS) inequalities [35, Theorems 4.3 and 8.3] that for \(\frac{N+\alpha }{N}\le p\le \frac{N+\alpha }{N-2}\) and \(2<q\le \frac{2N}{N-2}\) the energy functional \(\mathcal {I}_{\varepsilon }\in C^1(H^1({\mathbb {R}}^N),{\mathbb {R}})\), cf. [42]. For \(p>\frac{N+\alpha }{N-2}\) the energy \(\mathcal {I}_{\varepsilon }\) is not well-defined on \(H^1({\mathbb {R}}^N)\). In this case an additional assumption \(q>\frac{2Np}{N+\alpha }\) ensures the control of the nonlocal term by the \(L^q\) and \(L^2\)-norm via the HLS inequality and interpolation, i.e.
for a \(\theta \in (0,1)\). As a consequence, for \(p>\frac{N+\alpha }{N-2}\) and \(q>\frac{2Np}{N+\alpha }\) the energy \(\mathcal {I}_{\varepsilon }\) is well-defined on the space
Clearly, \({\mathcal {H}_q}\) endowed with the norm
is a Banach space, \({\mathcal {H}_q}\hookrightarrow L^{\frac{2Np}{N+\alpha }}({\mathbb {R}}^{N})\) for any \(q>2\) and \({\mathcal {H}_q}=H^1({\mathbb {R}}^N)\) when \(2<q\le 2^*\). It is easy to check that \(\mathcal {I}_{\varepsilon }\in C^1({\mathcal {H}_q}, {\mathbb {R}})\) and the problem (\(P_{\varepsilon }\)) is variationaly well-posed, in the sense that weak solutions \(u\in {\mathcal {H}_q}\) of (\(P_{\varepsilon }\)) are critical points of \(\mathcal {I}_{\varepsilon }\), i.e.
for all \(\varphi \in {\mathcal {H}_q}\). In particular, weak solutions \(u\in {\mathcal {H}_q}\) of (\(P_{\varepsilon }\)) satisfy the Nehari identity
It is standard to see that under minor regularity assumptions weak solutions of (\(P_{\varepsilon }\)) also satisfy the Pohožaev identity.
Proposition 4.1
(Pohožaev identity) Assume \(p>1\) and \(q>2\). Let \(u\in {\mathcal {H}_q}\cap L^\frac{2Np}{N+\alpha }({\mathbb {R}}^N)\) be a weak solution of (\(P_{\varepsilon }\)). If \(\nabla u\in L^{\frac{2Np}{N+\alpha }}_{loc}({\mathbb {R}}^N)\cap H^{1}_{loc}({\mathbb {R}}^N)\) then
Proof
The proof is an adaptation of [43, Proposition 3.1], we omit the details. \(\square \)
As a consequence, we conclude that the existence range stated in Theorem 1.1 is optimal.
Corollary 4.1
(Nonexistence) Let \(1<p\le \frac{N+\alpha }{N}\) and \(q>2\), or \(p\ge \frac{N+\alpha }{N-2}\) and \(2<q\le \frac{2Np}{N+\alpha }\). Then (\(P_{\varepsilon }\)) has no nontrivial weak solutions \(u\in {\mathcal {H}_q}\cap L^\frac{2Np}{N+\alpha }({\mathbb {R}}^N)\cap W^{1,\frac{2Np}{N+\alpha }}_{loc}({\mathbb {R}}^N)\cap W^{2,2}_{loc}({\mathbb {R}}^N)\).
Proof
Follows from Pohožaev and Nehari identities. \(\square \)
4.2 Apriori regularity and decay at infinity
We show that all weak nonnegative solutions of (\(P_{\varepsilon }\)) are in fact bounded classical solutions with an \(L^1\)-decay at infinity. We first prove a partial results which relies on the maximum principle for the Laplacian.
Lemma 4.1
Assume \(p>\frac{N+2}{N}\) and \(q>\max \{p,2\}\). Let \(s>\frac{Np}{\alpha }\) and \(u\in {\mathcal {H}_q}\cap L^s({\mathbb {R}}^N)\) be a nonnegative weak solution of (\(P_{\varepsilon }\)). Then \(u\in L^\infty ({\mathbb {R}}^N)\).
Proof
The assumption \(s>\frac{Np}{\alpha }\) imply that \(I_{\alpha }*|u|^p\in L^\infty ({\mathbb {R}}^N)\). Then \(u\ge 0\) weakly satisfies
where \(C=C(u)=\Vert I_{\alpha }*|u|^p\Vert _\infty \). Choose \(m=m(u)>0\) such that \(Cm^{p-1}-m^{q-1}=0\). Testing against \(u_m=(u-m)_+\in H^1({\mathbb {R}}^N)\), we obtain
so \(\Vert u\Vert _\infty \le m\). \(\square \)
The proof of the next statement in the case \(p<\frac{N+\alpha }{N-2}\) is an adaptation of the iteration arguments in [43, Proposition 4.1]. We only outline the main steps of the proof. The case \(p\ge \frac{N+\alpha }{N-2}\) is new and relies heavily on the contraction inequality (A.3), which is discussed in the appendix.
Proposition 4.2
(Regularity and positivity) Let \(\frac{N+\alpha }{N}<p<\frac{N+\alpha }{N-2}\) and \(q>2\), or \(p\ge \frac{N+\alpha }{N-2}\) and \(q>\frac{2Np}{N+\alpha }\). If \(0\le u\in {\mathcal {H}_q}\) is a nontrivial weak solution of (\(P_{\varepsilon }\)) then \(u\in L^1\cap C^2({\mathbb {R}}^N)\) and \(u(x)>0\) for all \(x\in {\mathbb {R}}^N\).
Proof
Since \(u\in {\mathcal {H}_q}\) we know that \(u\in L^s({\mathbb {R}}^N)\) for all \(s\in [2,q^*]\). \(\square \)
Step 1. \(u\in L^1({\mathbb {R}}^N)\).
Proof
Note that \(u\ge 0\) weakly satisfies the inequality
Since \((-\Delta +{\varepsilon })^{-1}:L^s({\mathbb {R}}^N)\mapsto L^s({\mathbb {R}}^N)\) is a bounded order preserving linear mapping for any \(s\ge 1\), we have
Then, by the HLS and Hölder inequalities, \(u\in L^{\underline{s}_n}({\mathbb {R}}^N)\) with \(0<\frac{2p-1}{\underline{s}_{n}}-\frac{\alpha }{N}<1\) implies \(u\in L^{\underline{s}_{n+1}}({\mathbb {R}}^N)\), where
Since \(p>\frac{N+\alpha }{N}\), we start the \(\underline{s}_n\)-iteration with \(\underline{s}_0=\frac{2Np}{N+\alpha }<\frac{2N(p-1)}{\alpha }\), as in [43]. Then we achieve \(\underline{s}_{n+1}\ge 1\) after a finite number of steps. This implies \(u\in L^1({\mathbb {R}}^N)\). \(\square \)
Step 2. \(u\in L^\infty ({\mathbb {R}}^N)\).
Proof
Assume that \(q\le \frac{Np}{\alpha }\), otherwise we conclude by Lemma 4.1. We consider separately the cases \(p<\frac{N+\alpha }{N-2}\) and \(p\ge \frac{N+\alpha }{N-2}\), which use different structures within the Eq. (\(P_{\varepsilon }\)).
A. Case \(p<\frac{N+\alpha }{N-2}\). Note that \(u\ge 0\) weakly satisfies the inequality
Since \(u\in L^1({\mathbb {R}}^N)\), we have \((I_{\alpha }*u^p)u^{p-1}\in L^t({\mathbb {R}}^N)\) with \(\frac{1}{t}:=2p-1-\frac{\alpha }{N}>\frac{2}{N}\). We conclude that
Then, by the HLS and Hölder inequalities, \(u\in L^{\overline{s}_n}({\mathbb {R}}^N)\) with \(0<\frac{2p-1}{\overline{s}_{n}}-\frac{\alpha +2}{N}<1\) implies \(u\in L^{\overline{s}_{n+1}}({\mathbb {R}}^N)\), where
Since \(p<\frac{N+\alpha }{N-2}\), we start the \(\overline{s}_n\)-iteration with \(\overline{s}_0=\frac{2Np}{N+\alpha }>\frac{2N(p-1)}{\alpha +2}\), as in [43]. Then we achieve \(\overline{s}_{n+1}>\frac{Np}{\alpha }\) after a finite number of steps. (Or if \(\overline{s}_{n+1}=\frac{Np}{\alpha }\) we readjust \(\overline{s}_0\).)
B. Case \(p\ge \frac{N+\alpha }{N-2}\) and \(q>\frac{2Np}{N+\alpha }\). Note that \(u\ge 0\) weakly satisfies the inequality
Then, by the HLS and Hölder inequalities, and by the contraction inequality A.3, \(u\in L^{\overline{s}_n}({\mathbb {R}}^N)\) with \(0<\frac{2p-1}{\overline{s}_{n}}-\frac{\alpha }{N}<1\) implies \(u\in L^{\overline{s}_{n+1}}({\mathbb {R}}^N)\), where
We start the \(\overline{s}_n\)-iteration with \(\overline{s}_0=q\). If \(q\ge 2p\) we achieve \(\overline{s}_{n+1}>\frac{Np}{\alpha }\) after a finite number of steps. If \(q<2p\) we note that since \(p\ge \frac{N+\alpha }{N-2}\), we have \(\overline{s}_0=q>\frac{2Np}{N+\alpha }\). Then we again achieve \(\overline{s}_{n+1}>\frac{Np}{\alpha }\) after a finite number of steps. (Or if \(\overline{s}_{n+1}=\frac{Np}{\alpha }\) we readjust \(\overline{s}_0\).) \(\square \)
Step 3. \(u\in W^{2,r}({\mathbb {R}}^N)\) for every \(r>1\) and \(u\in C^2({\mathbb {R}}^N)\).
Proof
Since \(u\in L^1\cap L^\infty ({\mathbb {R}}^N)\), we have
where \(F:=(I_{\alpha }*u^p)u^{p-1}-u^{q-1}\in L^\infty ({\mathbb {R}}^N)\). Then the conclusion follows by the standard Schauder estimates, see [43, p. 168] for details. \(\square \)
Step 4. \(u(x)>0\) for all \(x\in {\mathbb {R}}^N\).
Proof
We simply note that \(u\ge 0\) satisfies
where \(V:={\varepsilon }+u^{q-2}\in C({\mathbb {R}}^N)\). Then \(u(x)>0\) for all \(x\in {\mathbb {R}}^N\), e.g. by the weak Harnack inequality. \(\square \)
Proposition 4.3
(Decay asymptotics) Let \(\frac{N+\alpha }{N}<p<\frac{N+\alpha }{N-2}\) and \(q>2\) or \(p\ge \frac{N+\alpha }{N-2}\) and \(q>\frac{2Np}{N+\alpha }\). Let \(0<u_{\varepsilon }\in L^1\cap C^2({\mathbb {R}}^N)\) be a radially symmetric solution of (\(P_{\varepsilon }\)). Then
Moreover, there exists \(C_{\varepsilon }>0\) such that
-
if \(p > 2\),
$$\begin{aligned} \lim _{|x | \rightarrow \infty } u_{\varepsilon }(x) |x |^{\frac{N - 1}{2}} e^{\sqrt{{\varepsilon }}|x |}=C_{\varepsilon }, \end{aligned}$$ -
if \(p = 2\),
$$\begin{aligned} \lim _{|x | \rightarrow \infty } u_{\varepsilon }(x) |x |^{\frac{N - 1}{2}} \exp \int _{\nu }^{|x |} \sqrt{{\varepsilon }- \tfrac{\nu ^{N - \alpha }}{s^{N - \alpha }}} \,ds=C_{\varepsilon },\quad \text {where}\quad \nu :=\left( A_\alpha \Vert u_{\varepsilon }\Vert _2^2\right) ^\frac{1}{N - \alpha }, \end{aligned}$$ -
if \(p < 2\),
$$\begin{aligned} \lim _{x\rightarrow \infty }u_{\varepsilon }(x)|x|^\frac{N-\alpha }{2-p}=\left( {\varepsilon }^{-1}A_\alpha \Vert u_{\varepsilon }\Vert _p^p\right) ^{\frac{1}{2-p}}. \end{aligned}$$
Proof
To simplify the notation, we drop the subscript \({\varepsilon }\) for \(u_{\varepsilon }\) in this proof.
Let \(u\in L^1\cap C^2({\mathbb {R}}^N)\) be a positive radially symmetric solution of (\(P_{\varepsilon }\)). By the Strauss’ radial \(L^1\)-bound [6, Lemma A.4], \(u^p(|x|)\le C\Vert u\Vert _1|x|^{-Np}\) for all \(x\in {\mathbb {R}}^N\). Then by [43, Propositions 6.1], there exists \(\mu \in {\mathbb {R}}\) such that
with \(0 < \delta \le \min (1, N (p - 1))\). In particular, this implies (4.11).
Case \(p\ge 2\). The derivation of the decay asymptotic of u in the case(s) \(p\ge 2\) requires minimal modifications of the proofs of [43, Propositions 6.3, 6.5]. Indeed, (4.11) implies \((I_\alpha *|u|^p)u^{p-2}(x)\le \frac{3}{4} {\varepsilon }\) for all \(|x|>\rho \), for some \(\rho >0\). Therefore, u satisfies
As in [43, Propositions 6.3] we conclude that
Therefore, u is a solution of
where
The initial rough upper bound (4.13) implies that the term \(u^{q-2}(x)\) in the linearisation potential \(W_{\varepsilon }(x)\) has an exponential decay and is negligible in the subsequent asymptotic analysis of Propositions 6.3 and 6.5 in [43]. We omit the details.
Case \(p< 2\). This proof is an adaptation of [43, Propositions 6.6].
To derive the upper bound, we note that by Young’s inequality,
By (4.12), we have
Therefore, u satisfies the inequality
Define now \(\bar{u} \in C^2 ({\mathbb {R}}^N \setminus B_1)\) by
We now apply [43, Lemma 6.7] twice and use the linearity of the operator \(-\Delta +{\varepsilon }(2-p)\) to obtain
By the comparison principle, we have \(u \le \bar{u}\) in \({\mathbb {R}}^N \setminus B_1\). Thus
To deduce the lower bound, note that by the chain rule, \(u^{2 - p} \in C^2 ({\mathbb {R}}^N)\) and
Since \(p \in (1, 2)\) and \(q>2\), by the equation satisfied by u and by (4.12) and (4.14), for some \(c_{\varepsilon }>0\) we have
Let \(\underline{u} \in C^2({\mathbb {R}}^N \setminus B_1)\) be such that
We apply now [43, Lemma 6.7] to deduce
By the comparison principle, \(\underline{u} \le u^{2 - p} \) in \({\mathbb {R}}^N \setminus B_1\). We conclude that
and the assertion follows from the combination of (4.15) and (4.14). \(\square \)
4.3 Proof of the existence
Throughout this section, we assume that either \(\frac{N+\alpha }{N}<p<\frac{N+\alpha }{N-2}\) and \(q>2\) or \(p\ge \frac{N+\alpha }{N-2}\) and \(q>\frac{2Np}{N+\alpha }\). We construct a groundstate of (\(P_{\varepsilon }\)) by minimising over the Pohožaev manifold of (\(P_{\varepsilon }\)). Similar approach for Choquard’s equations with different classes of nonlinearities was recently used in [25, 32].
Set
where \(\mathcal {P}_{{\varepsilon }}:{\mathcal {H}_q}\rightarrow {\mathbb {R}}\) is defined by
For each \(u\in {\mathcal {H}_q}\setminus \{0\}\), set
Then
Clearly, there exists a unique \(t_u>0\) such that \(f_u(t_u)=\max \{f_u(t): t>0\}\) and \(f'_u(t_u)t_u=0\), which means that \(u(x/t_u)\in \mathscr {P}_{{\varepsilon }}\). Therefore \(\mathscr {P}_{{\varepsilon }}\ne \emptyset \).
Define \(M: {\mathcal {H}_q}\rightarrow {\mathbb {R}}\) as
Then \(M(u)=0\) if and only if \(u=0\). Taking into account the definition of M and the norm \(\Vert \cdot \Vert _{\mathcal {H}_q}\), we can check that
where \(C>0\) is independent of \(u\in {\mathcal {H}_q}\).
Lemma 4.2
Assume that either \(\frac{N+\alpha }{N}<p<\frac{N+\alpha }{N-2}\) and \(q>2\), or \(p\ge \frac{N+\alpha }{N-2}\) and \(q>\frac{2Np}{N+\alpha }\). Then there exists \(C>0\) such that for all \(u\in {\mathcal {H}_q}\),
Proof
For each \(u\in {\mathcal {H}_q}\setminus \{0\}\), let \(u_t\) be defined in (4.16). If \(M(u)\le 1\) then we set \(t=M(u)^{-\frac{1}{N}}\ge 1\), and we have \(M(u_t)\le t^NM(u)=1\). Thus it follows from the HLS inequality, the embedding \({\mathcal {H}_q}\hookrightarrow L^{\frac{2Np}{N+\alpha }}({\mathbb {R}}^{N})\), and (4.19) that
To clarify the last inequality, note that \(M(u_t)\le t^N M(u)=1\). Then using (4.17), we obtain
If \(M(u)> 1\) then we set \(t=M(u)^{-\frac{1}{N-2}}<1\), and we have \(M(u_t)\le t^{N-2}M(u)=1\). Similarly as before, we conclude that
which completes the proof. \(\square \)
To find a groundstate solution of (\(P_{\varepsilon }\)), we prove the existence of a spherically symmetric nontrivial nonegative minimizer of the minimization problem
and then show that \(\mathscr {P}_{{\varepsilon }}\) is a natural constraint for \(\mathcal {I}_{\varepsilon }\), i.e. the minimizer \(u_0\in \mathscr {P}_{{\varepsilon }}\) satisfies \(\mathcal {I}_{\varepsilon }^\prime (u_0)=0\). Such approach for the local equations goes back at least to [50] in the local case and to [48] in the case of nonlocal problems.
We divide the proof of the existence of the groundstate into several steps.
Step 1. \(0\notin \partial \mathscr {P}_{{\varepsilon }}\).
Proof
Indeed, for \(u\in \mathscr {P}_{{\varepsilon }}\), we have, by using the HLS and Sobolev inequalities,
which means that there exists \(C>0\) such that \(M(u)\ge C\) for all \(u\in \mathscr {P}_{{\varepsilon }}\). \(\square \)
Step 2. \(c_{{\varepsilon }}=\inf _{u\in \mathscr {P}_{{\varepsilon }}} \mathcal {I}_{{\varepsilon }}(u)>0\)
Proof
For each \(u\in \mathscr {P}_{{\varepsilon }}\), we have
therefore \(c_{{\varepsilon }}\ge 0\). If \(c_{{\varepsilon }}=0\), then there exists a sequence \(\{u_n\}\subset \mathscr {P}_{{\varepsilon }}\) such that \(\mathcal {I}_{{\varepsilon }}(u_n)\rightarrow 0\), which means that \(\Vert \nabla u_n\Vert _2^2\rightarrow 0\) and \(\int _{{\mathbb {R}}^N}(I_{\alpha }*|u_n|^p)|u_n|^pdx\rightarrow 0\). Recall that \(\mathcal {P}_{{\varepsilon }}(u_n)=0\). Then we conclude that \(\Vert u_n\Vert _2^2\rightarrow 0\) and \(\Vert u_n\Vert ^q_q\rightarrow 0\). This implies that \(\Vert u_n\Vert _{\mathcal {H}_q}^2\rightarrow 0\), which contradicts to \(0\notin \partial \mathscr {P}_{{\varepsilon }}\). \(\square \)
Step 3. There exists \(u_0\in \mathscr {P}_{{\varepsilon }}\) such that \(\mathcal {I}_{{\varepsilon }}(u_0)=c_{\varepsilon }\).
Proof
Since \(c_{\varepsilon }\) is well defined, there exists a sequence \(\{u_n\}\subset \mathscr {P}_{{\varepsilon }}\) such that \(\mathcal {I}_{{\varepsilon }}(u_n)\rightarrow c_{\varepsilon }\). It follows from (4.21) that both \(\{\Vert \nabla u_n\Vert _2^2\}\) and \(\{\int _{{\mathbb {R}}^N}(I_{\alpha }*|u_n|^p)|u_n|^pdx\}\) are bounded. Note that \(\mathcal {P}_{{\varepsilon }}(u_n)=0\). Then we see that \(\{\Vert u_n\Vert _2^2\}\) and \(\{\Vert u_n\Vert _q^q\}\) are bounded, and therefore \(\{u_n\}\) is bounded in \({\mathcal {H}_q}\).
Let \(u^*_n\) be the Schwartz spherical rearrangement of \(|u_n|\). Then \(u_n^*\in {\mathcal {H}_{q,rad}}\), the subspace of \({\mathcal {H}_q}\) which consists of all spherically symmetric functions in \({\mathcal {H}_q}\), and
cf. [35, Section 3]. For each \(u^*_n\), there exists a unique \(t_n\in (0, 1)\) such that \(v_n:=u^*_n(\frac{x}{t_n})\in \mathscr {P}_{{\varepsilon }}\). Therefore we obtain that
which implies that \(\{v_n\}\) is also a minimizing sequence for \(c_{{\varepsilon }}\), that is \(\mathcal {I}_{{\varepsilon }}(v_n)\rightarrow c_{\varepsilon }\). (In fact, we can also prove that \(t_n\rightarrow 1\).)
Clearly \(\{v_n\}\subset {\mathcal {H}_{q,rad}}\) is bounded. Then there exists \(v\in {\mathcal {H}_{q,rad}}\) such that \(v_n\rightharpoonup v\) weakly in \({\mathcal {H}_q}\) and \(v_n(x)\rightarrow v(x)\) for a.e. \(x\in {\mathbb {R}}^N\), by the local compactness of the emebedding \({\mathcal {H}_q}\hookrightarrow L^2_{loc}({\mathbb {R}}^N)\) on bounded domains. Using Strauss’s \(L^s\)-bounds with \(s=2\) and \(s=q^*\), we conclude that
Since \(U\in L^s({\mathbb {R}}^N)\) for \(s\in (2,q^*)\), by the Lebesgue dominated convergence we conclude that for \(s\in (2, q^*)\),
Note that \(q^*>\frac{2Np}{N+\alpha }\) and hence we can always choose \(s>\frac{2Np}{N+\alpha }>p\) such that \(\{v_n\}\) is bounded in \(L^s({\mathbb {R}}^N)\). Then by the nonlocal Brezis–Lieb Lemma with high local integrability [40, Proposition 4.7] we conclude that
This means that \(v\ne 0\), since by Lemma 4.2 the sequence \(\{M(v_n)\}\) has a positive lower bound. Then there exists a unique \(t_0>0\) such that \(v(\frac{x}{t_0})\in \mathscr {P}_{{\varepsilon }}\). By the weakly lower semi-continuity of the norm, we see that
which implies that \(\mathcal {I}_{{\varepsilon }}(v(\frac{x}{t_0}))=c_{\varepsilon }\). We conclude this step by taking \(u_0(x):=v(\frac{x}{t_0})\). \(\square \)
Step 4. \(\mathcal {P}_{\varepsilon }^\prime (u_0)\ne 0\), where \(u_0\) is obtained in Step 3.
Proof
Arguing by contradiction, we assume that \(\mathcal {P}_{\varepsilon }^\prime (u_0)=0\). Then \(u_0\) is a weak solution of the following equation,
By Propositions 4.1 and 4.2, \(u_0\) satisfies the Pohožaev identity
This, together with \(\mathcal {P}_{{\varepsilon }}(u_0)=0\), implies that
which contradict \(u_0\ne 0\). \(\square \)
Step 5. \(\mathcal {I}_{{\varepsilon }}'(u_0)=0\), i.e., \(u_0\) is a weak solution of (\(P_{\varepsilon }\)).
Proof
By the Lagrange multiplier rule, there exists \(\mu \in {\mathbb {R}}\) such that \(\mathcal {I}_{{\varepsilon }}'(u_0)=\mu \mathcal {P}_{{\varepsilon }}'(u_0)\). We claim that \(\mu =0\). Indeed, since \(\mathcal {I}_{{\varepsilon }}'(u_0)=\mu \mathcal {P}_{{\varepsilon }}'(u_0)\), then \(u_0\) satisfies in the weak sense the following equation,
By Propositions 4.1 and 4.2, \(u_0\) satisfies the Pohožaev identity
By using \(\mathcal {P}_{{\varepsilon }}(u_0)=0\) again, we conclude that
which means that \(\mu =0\). Therefore \(\mathcal {I}_{{\varepsilon }}'(u_0)=0\). \(\square \)
5 Existence and properties of groundstates for (\(P_0\))
In this section we study the existence and some qualitative properties of groundstate solutions for the equation
where \(N\ge 3\), \(\alpha \in (0,N)\), \(p>1\) and \(q>2\). Equation (\(P_0\)) appears as a formal limit of (\(P_{\varepsilon }\)) with \({\varepsilon }=0\). The natural domain for the formal limit energy \(\mathcal {I}_0\) which corresponds to (\(P_0\)) is the space
Clearly, \({\mathcal {D}_q}\) endowed with the norm
is a Banach space, and \({\mathcal {D}_q}\hookrightarrow L^q\cap L^{2^*}({\mathbb {R}}^{N})\). In particular, \({\mathcal {D}_q}\hookrightarrow L^{\frac{2Np}{N+\alpha }}({\mathbb {R}}^{N})\) for \(\frac{N+\alpha }{N}<p<\frac{N+\alpha }{N-2}\) and \(2<q<\frac{2Np}{N+\alpha }\), or \(p\ge \frac{N+\alpha }{N-2}\) and \(q>\frac{2Np}{N+\alpha }\). However, \({\mathcal {H}_q}\subsetneq {\mathcal {D}_q}\). Hence (\(P_{\varepsilon }\)) can not be considered as a small perturbation of (\(P_0\)), since the domain of \(\mathcal {I}_0\) is strictly bigger than the domain of \(\mathcal {I}_{\varepsilon }\).
If \(p\in (\frac{N+\alpha }{N}, \frac{N+\alpha }{N-2})\) and \(q\in (2, \frac{2Np}{N+\alpha })\) or \(p\ge \frac{N+\alpha }{N-2}\) and \(q>\frac{2Np}{N+\alpha }\) then the HLS, Sobolev and interpolation inequalities ensure the control of the nonlocal term by the \(L^q\) and \(D^1\)-norms,
with a \(\theta \in (0,1)\). Then it is standard to check that \(\mathcal {I}_0\in C^1({\mathcal {D}_q}, {\mathbb {R}})\) and the problem (\(P_0\)) is variationaly well-posed, in the sense that weak solutions \(u\in {\mathcal {D}_q}\) of (\(P_{\varepsilon }\)) are critical points of \(\mathcal {I}_{\varepsilon }\), i.e.
for all \(\varphi \in {\mathcal {D}_q}\). In particular, weak solutions \(u\in {\mathcal {D}_q}\) of (\(P_0\)) satisfy the Nehari identity
As in Proposition 4.1, we see that weak solutions \(u\in {\mathcal {D}_q}\cap L^\frac{2Np}{N+\alpha }({\mathbb {R}}^N)\cap W^{1,\frac{2Np}{N+\alpha }}_{loc}({\mathbb {R}}^N)\cap W^{2,2}_{loc}({\mathbb {R}}^N)\) of (\(P_0\)) also satisfy the Pohožaev identity
We are going to prove the existence of a ground state of \((P_0)\) by minimizing over the Pohožaev manifold \(\mathcal {P}_0\). This requires apriori additional regularity and some decay properties of the weak solutions.
Lemma 5.1
(\(L^1\)-decay) Let \(\frac{N+\alpha }{N}<p<\frac{N+\alpha }{N-2}\) and \(2<q<\frac{2Np}{N+\alpha }\). If \(0\le u\in {\mathcal {D}_q}\) is a weak solution of (\(P_0\)) then \(u\in L^1({\mathbb {R}}^N)\).
Proof
Note that \(u\ge 0\) weakly satisfies the inequality (4.9). Then, by the HLS and Hölder inequalities, and by the contraction inequality A.3, \(u\in L^{\underline{s}_n}({\mathbb {R}}^N)\) with \(0<\frac{2p-1}{\underline{s}_{n}}-\frac{\alpha }{N}<1\) implies \(u\in L^{\underline{s}_{n+1}}({\mathbb {R}}^N)\), where
We start the \(\underline{s}_n\)-iteration with \(\underline{s}_0=q<\frac{2Np}{N+\alpha }\). Then we achieve \(\underline{s}_{n+1}\le 1\) after a finite number of steps. \(\square \)
Proposition 5.1
(Regularity) Let \(\frac{N+\alpha }{N}<p<\frac{N+\alpha }{N-2}\) and \(2<q<\frac{2Np}{N+\alpha }\), or \(p\ge \frac{N+\alpha }{N-2}\) and \(q>\frac{2Np}{N+\alpha }\). If \(0\le u\in {\mathcal {D}_q}\) is a nontrivial weak solution of (\(P_0\)) then \(u\in C^2({\mathbb {R}}^N)\) and \(u(x)>0\) for all \(x\in {\mathbb {R}}^N\).
Proof
Since \(0\le u\in {\mathcal {D}_q}\) we know that \(u\in L^q\cap L^{2^*}({\mathbb {R}}^N)\). Assume that \(q\le \frac{Np}{\alpha }\), otherwise we conclude that \(u\in L^\infty ({\mathbb {R}}^N)\) by a modification of the comparison argument of Lemma 4.1.
If \(p>\frac{N+\alpha }{N-2}\) and \(q>\frac{2Np}{N+\alpha }\) we can show that \(u\in L^\infty ({\mathbb {R}}^N)\) by repeating the same iteration argument as in the proof of Proposition 4.2, Step 2(B).
If \(\frac{N+\alpha }{N}<p<\frac{N+\alpha }{N-2}\) and \(2<q<\frac{2Np}{N+\alpha }\) we know additionally that \(u\in L^1\cap L^{2^*}({\mathbb {R}}^N)\) by Lemma 5.1. Then we can conclude that \(u\in L^\infty ({\mathbb {R}}^N)\) by repeating the iteration argument in the proof of Proposition 4.2, Step 2(A).
Finally, \(u\in L^{q^*}\cap L^\infty ({\mathbb {R}}^N)\) implies \(u\in C^2({\mathbb {R}}^N)\) by the standard Hölder and Schauder estimates, while positivity of u(x) follows via the weak Harnack inequality, as in the proof of Proposition 4.2, Steps 3 and 4. \(\square \)
Unlike in the case \({\varepsilon }>0\), for \(p>\frac{N+\alpha }{N-2}\) we can not conclude that \(u\in L^1({\mathbb {R}}^N)\) via a regularity type iteration arguments. In fact, the decay of groundstates of (\(P_0\)) is more complex.
Proposition 5.2
(Decay estimates) Assume that either \(p<\frac{N+\alpha }{N-2}\) and \(q<\frac{2Np}{N+\alpha }\) or \(p>\frac{N+\alpha }{N-2}\) and \(q>\frac{2Np}{N+\alpha }\). Let \(u\in {\mathcal {D}_{q,rad}}\) be a nontrivial nonnegative weak solution of (\(P_0\)). Then:
-
if \(p < \frac{N+\alpha }{N-2}\) then \(u\in L^1({\mathbb {R}}^N)\),
-
if \(p > \frac{N+\alpha }{N-2}\) then
$$\begin{aligned} u_0 > rsim |x|^{-(N-2)}\quad \text {as}\quad |x|\rightarrow \infty , \end{aligned}$$and if \(p>\max \left\{ \frac{N+\alpha }{N-2},\frac{2}{3}\left( 1+\frac{N+\alpha }{N-2}\right) \right\} \) then
$$\begin{aligned} u_0\sim |x|^{-(N-2)}\quad \text {as}\quad |x|\rightarrow \infty . \end{aligned}$$(5.4)
Proof
The case \(p<\frac{N+\alpha }{N-2}\) is the content of Lemma 5.1.
Assume \(p>\frac{N+\alpha }{N-2}\). Then \(u\in L^{2^*}\cap C^2({\mathbb {R}}^N)\). By the Strauss’s radial \(L^s\)-bounds with \(s=2^*\),
Since \(q>2^*\), \(u>0\) satisfies
where
for some \(\delta >0\). By comparing with an explicit subsolution \(c|x|^{-(N-2)}(1+|x|^{-\delta /2})\) in \(|x|>1\), we conclude that
Assume now that \(p\ge \max \left\{ \frac{N+\alpha }{N-2},2\right\} \). Using again (5.5) and we conclude that
for some \(\delta \in (0,1)\). Therefore, \(u>0\) satisfies
By comparing with an explicit supersolution \(c|x|^{-(N-2)}(1-|x|^{-\delta /2})\) in \(|x|>2\), we conclude that
Next we assume that \(\alpha <N-4\) and \(\frac{N+\alpha }{N-2}<p<2\). Using again (5.5), the lower bound (5.6) and taking into account that \(p<2\), we conclude that
where \(\delta >0\) provided that \(p>\frac{2}{3}\left( 1+\frac{N+\alpha }{N-2}\right) \). Then we conclude as before. \(\square \)
Proof of Theorem 2.1
We assume that either \(\frac{N+\alpha }{N}<p<\frac{N+\alpha }{N-2}\) and \(2<q<\frac{2Np}{N+\alpha }\) or \(p>\frac{N+\alpha }{N-2}\) and \(q>\frac{2Np}{N+\alpha }\). Set
where \(\mathcal {P}_{0}:{\mathcal {D}_q}\rightarrow {\mathbb {R}}\) is defined by
As in the case \({\varepsilon }>0\), it is standard to check that \(\mathscr {P}_{0}\ne \emptyset \) (see (4.17)). To construct a groundstate solution of \((P_0)\), we prove the existence of a spherically symmetric nontrivial nonegative minimizer of the minimization problem
and then show that \(\mathscr {P}_{0}\) is a natural constraint for \(\mathcal {I}_0\), i.e. the minimizer \(u_0\in \mathscr {P}_{0}\) satisfies \(\mathcal {I}_0^\prime (u_0)=0\). The arguments follow closely the proof of Theorem 1.1, except that instead of the quantity M defined in (4.18), we use \(\overline{M}: {\mathcal {D}_q}\rightarrow {\mathbb {R}}\) defined by
It is easy check that
where \(C>0\) is independent of \(u\in {\mathcal {D}_q}\). Similarly to Lemma 4.2, we also can prove that there exists \(C>0\) such that for all \(u\in {\mathcal {D}_q}\),
which allows to control the nonlocal term. The remaining arguments follow closely Steps 1–5 in the proof of Theorem 1.1. We omit further details. \(\square \)
Remark 5.1
An equivalent route to construct a groundstate solution of (\(P_0\)) is to prove the existence of a minimizer of the problem
This does not require apriori regularity or decay properties of the weak solutions. It is standard but technical to establish the relation
and to prove that the minimization problems for \(a_0\) and \(c_0\) are equivalent up to a rescaling. Moreover, if \(w_0\in {\mathcal {D}_q}\) is a minimizer for \(a_0\) then
where \(\mu >0\) and \( u_0(x)=\mu ^{\frac{2}{4(p-1)-(\alpha +2)(q-2)}}w_0\left( \mu ^{\frac{q-2}{4(p-1)-(\alpha +2)(q-2)}} x\right) \) is a solution of (\(P_0\)).
Remark 5.2
Combining (5.2) and (5.3), we conclude that
which implies that (\(P_0\)) has no nontrivial solutions \(u\in {\mathcal {D}_q}\cap L^\frac{2Np}{N+\alpha }({\mathbb {R}}^N)\cap W^{1,\frac{2Np}{N+\alpha }}_{loc}({\mathbb {R}}^N)\cap W^{2,2}_{loc}({\mathbb {R}}^N)\) either if \(p<\frac{N+\alpha }{N-2}\) and \(q\ge \frac{2Np}{N+\alpha }\) or if \(p>\frac{N+\alpha }{N-2}\) and \(q\le \frac{2Np}{N+\alpha }\). Moreover, if \(p=\frac{N+\alpha }{N-2}\) then (\(P_0\)) has no nontrivial solution for \(q\ne \frac{2Np}{N+\alpha }\). This confirms that the existence assumptions of Theorem 2.1 on p and q are optimal, with one exception of the double-critical case \(p=\frac{N+\alpha }{N-2}\) and \(q=\frac{2Np}{N+\alpha }\).
Remark 5.3
In the double critical case \(p=\frac{N+\alpha }{N-2}\) and \(q=\frac{2Np}{N+\alpha }\) the Pohožaev argument does not lead to the nonexistence. In fact, it is not difficult to check (cf. [21, Lemma 1.1]) that the Emden–Fowler solution \(U_*\) defined in (2.9) satisfies
and the “Lagrange multiplier” can not be scaled out due to the scale invariance of the equation. It is an interesting open problem to show that a rescaling of \(U_*\) is a minimizer of the variational problem (5.11) in the double-critical case.
6 The Thomas–Fermi groundstate
To simplify notation we set in this section \(m:=q/2\). Denote
and
We first establish the following.
Proposition 6.1
Let \(m>\frac{2N}{N+\alpha }\). Then the minimization problem
admits a nonnegative spherically-symmetric nonincreasing minimizer \(\rho _*\in L^1\cap L^m({\mathbb {R}}^N)\).
Proof
Let \(\{\rho _n\}\subset \mathcal {A}_1\) be a minimizing sequence for \(s_{T\!F}\). Let \(\{\rho _n^*\}\) be the sequence of symmetric-decreasing rearrangements of \(\{\rho _n\}\). Then
see [35, Section 3]. Set \(\bar{\rho }_n^*(x):=\rho _n^*\left( (D_{\alpha }(\rho _n^*))^{\frac{1}{N+\alpha }}x\right) \). Then \(\bar{\rho }_n^*\in \mathcal {A}_1\) and as \(n\rightarrow \infty \),
which means that \(\{\bar{\rho }_n^*\} \subset \mathcal {A}_1\) is a minimizing sequence for \(s_{T\!F}\). Moreover, \(\{\bar{\rho }_n^*\}\) is bounded in \(L^1({\mathbb {R}}^N)\) and \(L^{m}({\mathbb {R}}^N)\). By Strauss’s lemma,
By Helly’s selection theorem for monotone functions, there exists a nonnegative spherically-symmetric nonincreasing function \(\rho _*(|x|)\le U(|x|)\) such that, up to a subsequence,
Since \(U\in L^s({\mathbb {R}}^N)\) for all \(s\in (1,m)\), by the Lebesgue’s dominated convergence we conclude that for every \(s\in (1, m)\),
Note that \(\frac{2N}{N+\alpha }\in (1, m)\). Then by the nonlocal Brezis–Lieb lemma [40, Proposition 4.7] we conclude that
which implies that \(0\ne \rho _*\in \mathcal {A}_1\). Therefore, using the standard Brezis–Lieb lemma,
that is, \(E(\rho _*)=s_{T\!F}\). Moreover, \(\bar{\rho }_n^*\rightarrow \rho _*\) strongly in \(L^1({\mathbb {R}}^N)\) and \(L^{m}({\mathbb {R}}^N)\). \(\square \)
Proof of Theorem 2.6
Let \(\rho _*\in L^1\cap L^m({\mathbb {R}}^N)\) be a minimizer for \(s_{T\!F}\), as constructed in Proposition 6.1. It is standard to show that \(\rho _*\) satisfies
for a Lagrange multiplier \(\lambda \in {\mathbb {R}}\). The proof can be adapted from [1] or [11, Proposition 3.6] and we omit it here. Since \(\rho _*\ge 0\), (6.2) is equivalent to the Thomas–Fermi equation
Testing (6.3) against \(\rho _*\) we conclude that
or taking into account the definition of \(s_{T\!F}\) we conclude that
To prove the virial identity (2.4) consider the rescaling \(\rho _t(x)=t^{-\frac{N+\alpha }{2}}\rho _*(x/t)\). Then
and
since \(m>\frac{2N}{N+\alpha }\) and hence \(t=1\) is the minimum of the differentiable function \(E(\rho _t)\), which implies that
Therefore we have
It follows from (6.3) that \(\rho _*\) satisfies
and the virial identity
To prove the \(L^\infty \)-bound on \(\rho _*\), in view of the Strauss’ radial bound (6.1) we only need to show that \(\rho _*\) is bounded near the origin. Observe that
If \(m>\frac{N}{\alpha }\) then \(I_\alpha *\rho _*\in C^{0,\alpha -\frac{N}{m}}({\mathbb {R}}^N)\) and hence \(\rho _*\in L^\infty ({\mathbb {R}}^N)\) in view of (6.3).
If \(m<\frac{N}{\alpha }\) we employ an \(L^s\)-iteration of the same structure as in our proof of Proposition 4.2, Step 2(B). Indeed, by the HLS and Hölder inequalities applied to (6.8), \(\rho _*\in L^{\overline{s}_n}({\mathbb {R}}^N)\) with \(0<\frac{1}{\overline{s}_{n}}-\frac{\alpha }{N}<1\) implies \(\rho _*\in L^{\overline{s}_{n+1}}({\mathbb {R}}^N)\), where
We start the \(\overline{s}_n\)-iteration with \(\overline{s}_0=m\). If \(m\ge 2\) we achieve \(\overline{s}_{n+1}>\frac{N}{\alpha }\) after a finite number of steps. If \(m<2\) we note that \(\overline{s}_0=m>\frac{2N}{N+\alpha }\) by the assumption. Then we again achieve \(\overline{s}_{n+1}>\frac{N}{\alpha }\) after a finite number of steps. (Or if \(\overline{s}_{n+1}=\frac{N}{\alpha }\) we readjust \(\overline{s}_0\).)
The compact support property is standard (cf. [11, Corollary 3.8]). We sketch the argument for completeness. Indeed, since \(\rho _*\in L^1({\mathbb {R}}^N)\) is a nonnegative radially nonincreasing function, it is known that for any \(\alpha \in (0,N)\),
(and \(C=\Vert \rho _*\Vert _1\) if \(\alpha >1\)), cf. [20, Corollary 2.3]. This is incompatible with (6.2), unless \(\rho _*\) has a compact support. Since \(\rho _*\) is nonincreasing we also conclude that \(\mathrm {supp}(\rho _*)\) is a connected set and hence must be a ball of radius \(R_*>0\) (and if \(\alpha >1\) then \(R_*\lesssim \Vert \rho _*\Vert _1^\frac{1}{N-\alpha }\)).
If \(\alpha >\left( \frac{m-2}{m-1}\right) _+\) then the Hölder regularity \(\rho _*\in C^{0,\gamma }({\mathbb {R}}^N)\) with \(\gamma =\min \{1,\frac{1}{m-1}\}\) follows exactly as in [12, Theorem 8]. We only note that the iteration steps (3.26), (3.27) in [12, p. 127] remain valid for any \(m\le 2\), as soon as \(\rho _*\in L^\infty ({\mathbb {R}}^N)\), which is ensured by our assumption \(m>\frac{2N}{N+\alpha }\). If \(m>2\) and \(\alpha \le \left( \frac{m-2}{m-1}\right) _+\) then \(\rho _*\in C^{0,\gamma }({\mathbb {R}}^N)\) for any \(\gamma <\frac{2}{m-2}\) by the same argument as in [12, Remark 2]. Further, \(\rho _*\in C^\infty (B_{R_*})\) can be deduced as in [12, Theorem 10].
Finally, keeping in mind that \(q=2m\), the function
is a groundstate of the Thomas–Fermi equation in the form
by direct scaling computation and in view of the properties of \(\rho _*\). \(\square \)
Remark 6.1
In [14, Proposition 5.16] the authors establish the existence of a unique bounded nonnegative radially nonincreasing solution to the Euler–Lagrange Eq. (6.7) in the range \(\alpha \in (0,2)\) and \(m\in \left( \frac{2N}{N+\alpha },m_c\right) \) (for \(\alpha =2\) the existence of a radial solution is classical, see e.g. [36, Theorem 5.1], while the uniqueness follows from [24, Lemma 5]). These existence results do not include an explicit variational characterisation of the solution in terms of \(s_{T\!F}\). However once the existence of a minimizer for \(s_{T\!F}\) is established (see Proposition 6.1), solutions constructed for \(\alpha \in (0,2]\) in [14, 36] coincide with the minimizer for \(s_{T\!F}\) in view of the uniqueness.
7 Asymptotic profiles: non-critical regimes
In this section we prove the convergence of rescaled groundstates \(u_{\varepsilon }\) to the limit profiles in the three noncritical regimes.
7.1 Formal limit \((P_0)\)
Throughout this section we assume that \(p>\frac{N+\alpha }{N-2}\) and \(q>\frac{2Np}{N+\alpha }\), or \(\frac{N+\alpha }{N}<p<\frac{N+\alpha }{N-2}\) and \(2<q<\frac{2Np}{N+\alpha }\).
Let \(u_{\varepsilon }\) be the positive spherically symmetric groundstate solution of (\(P_{\varepsilon }\)) constructed in Theorem 1.1, and \(c_{\varepsilon }=\mathcal {I}_{\varepsilon }(u_{\varepsilon })>0\) denotes the corresponding energy level, defined in (4.20). We are going to show that \(u_{\varepsilon }\) converges to the constructed in Theorem 2.1 positive spherically symmetric groundstate \(u_0\) of the formal limit Eq. (\(P_0\)), which has the energy \(c_0=\mathcal {I}_0(u_0)>0\), as defined in (\(\mathscr {C}_{HL}\)).
Below we present the proof only in the supercritical case \(p>\frac{N+\alpha }{N-2}\) and \(q>\frac{2Np}{N+\alpha }\). The subcritical case \(\frac{N+\alpha }{N}<p<\frac{N+\alpha }{N-2}\) and \(2<q<\frac{2Np}{N+\alpha }\) follows the same lines but easier, because in this case \(u_0\in L^1({\mathbb {R}}^N)\). The proof in the supercritical case relies on the decay estimate (5.4), which needs an additional restriction \(p>\max \left\{ \frac{N+\alpha }{N-2},\frac{2}{3}\left( 1+\frac{N+\alpha }{N-2}\right) \right\} \).
Lemma 7.1
Let \(p>\max \left\{ \frac{N+\alpha }{N-2},\frac{2}{3}\left( 1+\frac{N+\alpha }{N-2}\right) \right\} \) and \(q>\frac{2Np}{N+\alpha }\). Then \(0<c_{{\varepsilon }}-c_0\rightarrow 0\) as \({\varepsilon }\rightarrow 0\).
Proof
First, we use \(u_{\varepsilon }\) with \({\varepsilon }>0\) as a test function for \(\mathcal {P}_0\). We obtain
Hence there exists a unique \(t_{{\varepsilon }}\in (0, 1)\) such that \(u_{{\varepsilon }}(x/t_{{\varepsilon }})\in \mathscr {P}_0\), and we have
which means \(c_0<c_{{\varepsilon }}\).
To show that \(c_{{\varepsilon }}\rightarrow c_0\) as \({\varepsilon }\rightarrow 0\) we shall use \(u_0\) as a test function for \(\mathcal {P}_{\varepsilon }\). According to (5.4), \(u_0\in L^2({\mathbb {R}}^N)\) iff \(N\ge 5\). The dimensions \(N=3,4\) require a separate consideration.
Case \(N\ge 5\). Since \(\mathcal {P}_{{\varepsilon }}(u_0)=\frac{{\varepsilon }}{2}\Vert u_0\Vert _2^2>0\), there exists \(\overline{t}_{{\varepsilon }}>1\) such that \(u_0(x/\overline{t}_{{\varepsilon }})\in \mathscr {P}_{{\varepsilon }}\), i.e.,
This, combined with (5.3) and \(u_0\in L^2({\mathbb {R}}^N)\), implies that
Therefore, \(\overline{t}_{{\varepsilon }}\rightarrow 1\) as \({\varepsilon }\rightarrow 0\). Moreover,
where \(C>0\) is independent of \({\varepsilon }\). Thus we have
This, together with (7.1), means that \(c_{{\varepsilon }}-c_0\rightarrow 0\) as \({\varepsilon }\rightarrow 0\).
To consider the case \(N=3, 4\), given \(R\gg 1\), we introduce a cut-off function \(\eta _{R}\in C_c^{\infty }({\mathbb {R}}^N)\) such that \(\eta _R(r)=1\) for \(|r|<R\), \(0<\eta _R(r)<1\) for \(R<|r|<2R\), \(\eta _R(r)=0\) for \(|r|\ge 2R\) and \(|\eta _R'(r)|\le R/2\). It is standard to compute (cf. [52, Theorem 2.1]),
We will use \(\eta _R u_0\) with a suitable choice of \(R=R({\varepsilon })\) as a family of test functions for \(\mathcal {P}_{\varepsilon }\).
Case \(N=4\). By (7.2), (7.5), (7.4) and (7.3), for \(R\gg 1\) we have
Set \(R={\varepsilon }^{-1}\). Then for each \({\varepsilon }>0\) small, there exists \(\widetilde{t}_{{\varepsilon }}>1\) such that \(\mathcal {P}_{{\varepsilon }}(\eta _{R}(x/\widetilde{t}_{{\varepsilon }})u_0(x/\widetilde{t}_{{\varepsilon }}))=0\). Similarly to the case \(N\ge 5\), we can show that \(\widetilde{t}_{{\varepsilon }}\rightarrow 1\) as \({\varepsilon }\rightarrow 0\) and
We conclude that \(c_{{\varepsilon }}\le c_0+C{\varepsilon }\).
Case \(N=3\). By (7.2), (7.5), (7.4) and (7.3), for \(R\gg 1\) we have
Set \(R={\varepsilon }^{-1/2}\). Then for each \({\varepsilon }>0\) small, there exists \(\widehat{t}_{{\varepsilon }}>1\) such that \(\mathcal {P}_{{\varepsilon }}(\eta _{R}(x/\widehat{t}_{{\varepsilon }})u_0(x/\widehat{t}_{{\varepsilon }}))=0\) and \(\widehat{t}_{{\varepsilon }}\rightarrow 1\) as \({\varepsilon }\rightarrow 0\). We conclude as in the previous case. \(\square \)
Corollary 7.1
Let \(p>\max \left\{ \frac{N+\alpha }{N-2},\frac{2}{3}\left( 1+\frac{N+\alpha }{N-2}\right) \right\} \) and \(q>\frac{2Np}{N+\alpha }\). Then the quantities
are uniformly bounded as \({\varepsilon }\rightarrow 0\).
Proof
From Pohožaev and Nehari identities for \(\mathcal {I}_{{\varepsilon }}\) and Lemma 7.1 we have
\(\square \)
Lemma 7.2
Let \(p>\max \left\{ \frac{N+\alpha }{N-2},\frac{2}{3}\left( 1+\frac{N+\alpha }{N-2}\right) \right\} \) and \(q>\frac{2Np}{N+\alpha }\). Then \({\varepsilon }\Vert u_{{\varepsilon }}\Vert _2^2\rightarrow 0\) as \({\varepsilon }\rightarrow 0\).
Proof
Lemma 7.1 implies that there exists a unique \(t_{{\varepsilon }}\in (0, 1)\) such that \(u_{{\varepsilon }}(x/t_{{\varepsilon }})\in \mathscr {P}_0\). Indeed, assume that \(t_{{\varepsilon }}\rightarrow t_0<1\) as \({\varepsilon }\rightarrow 0\). Then by (7.1) we have, as \({\varepsilon }\rightarrow 0\),
which is a contradiction. Therefore \(t_{{\varepsilon }}\rightarrow 1\) as \({\varepsilon }\rightarrow 0\). Using \(\mathcal {P}_0(u_{{\varepsilon }}(x/t_{{\varepsilon }}))=0\) again, we see that
This implies that \({\varepsilon }\Vert u_{{\varepsilon }}\Vert _2^2\rightarrow 0\) as \({\varepsilon }\rightarrow 0\), since \(\mathcal {P}_{{\varepsilon }}(u_{{\varepsilon }})=0\).
Proof of Theorem 2.2
(case \(p>\max \left\{ \frac{N+\alpha }{N-2},\frac{2}{3}\left( 1+\frac{N+\alpha }{N-2}\right) \right\} \) and \(q>\frac{2Np}{N+\alpha }\)). From Corollary 7.1 and (7.6), we see that \(\{u_{{\varepsilon }}(x/t_{{\varepsilon }})\}\) is a minimizing sequence for \(c_0\) which is bounded in \(D^1_{rad}\cap L^q({\mathbb {R}}^N)\). Then there exists \(\overline{w}_0\in D^{1}_{rad}\cap L^q({\mathbb {R}}^N)\) such that
by the local compactness of the emebedding \(D^1({\mathbb {R}}^N)\hookrightarrow L^2_{loc}({\mathbb {R}}^N)\) on bounded domains. Using Strauss’ radial \(L^s\)-bounds with \(s=2^*\) and \(s=q\), we conclude that
Similarly to Step 3 in Sect. 4, using Lebesgue dominated convergence and nonlocal Brezis–Lieb Lemma [40, Proposition 4.7], we can show that \(u_{{\varepsilon }}(x/t_{{\varepsilon }})\rightarrow \overline{w}_0\) in \(D^{1}\cap L^q({\mathbb {R}}^N)\) and \(\overline{w}_0\) is a groundstate solution of \((P_0)\). \(\square \)
7.2 1st rescaling: Choquard limit
Throughout this section we assume that \(\frac{N+\alpha }{N}<p<\frac{N+\alpha }{N-2}\) and \(q>2\frac{2p+\alpha }{2+\alpha }\). The energy corresponding to the 1st rescaling
is given by
and the corresponding Pohožaev functional is
Note that \(\mathcal {I}_{\varepsilon }(u)={\varepsilon }^\frac{(N+\alpha )-p(N-2)}{2(p-1)}\mathcal {I}_{\varepsilon }^{(1)}(v)\) and consider the rescaled minimization problem
where \(\mathscr {P}_{{\varepsilon }}^{(1)}:=\{v\in H^1({\mathbb {R}}^N)\setminus \{0\}: \mathcal {P}_{{\varepsilon }}^{(1)}(v)=0\}\) is the Pohžaev manifold of (\(\mathscr {C}_{\varepsilon }\)). When \({\varepsilon }=0\), we formally obtain
It is known that \(c_0^{(1)}>0\) and \(c_0^{(1)}\) admits a minimizer \(v_0\in H^1({\mathbb {R}}^N)\), which is a groundstate of the Choquard Eq. (\(\mathscr {C}\)) characterised in Theorem 2.4 (see [43, Propositions 2.1 and 2.2]).
Let \(u_{\varepsilon }\) be the positive spherically symmetric groundstate solution of (\(P_{\varepsilon }\)) constructed in Theorem 1.1. Then the rescaled groundstate
is a groundstate solution of (\(\mathscr {C}_{\varepsilon }\)), i.e. \(\mathcal {I}_{\varepsilon }^{(1)}(v_{\varepsilon })=c_{\varepsilon }^{(1)}\). We are going to show that \(v_{\varepsilon }\) converges to the groundstate \(v_0\) of the Choquard Eq. (\(\mathscr {C}\)).
Lemma 7.3
\(0<c_{{\varepsilon }}^{(1)}-c_0^{(1)}\rightarrow 0\) as \({\varepsilon }\rightarrow 0\).
Proof
Clearly, \(\mathcal {P}_{{\varepsilon }}^{(1)}(v_{{\varepsilon }})=0\) implies that \(\mathcal {P}_{0}^{(1)}(v_{{\varepsilon }})<0\). Let \(w_{{\varepsilon }, t}(x)=v_{{\varepsilon }}(\frac{x}{t})\), then
has a unique maximum and \(\mathcal {P}_{0}^{(1)}(w_{{\varepsilon }, 1})<0\), thus there exists \(t_{v_{{\varepsilon }}}\in (0,1)\) such that \(w_{{\varepsilon }, t_{v_{{\varepsilon }}}}\in \mathscr {P}_{0}^{(1)}\). Therefore we have
On the other hand, let \(v_0\in \mathscr {P}^{(1)}_0\) be a radially symmetric ground state of (\(\mathscr {C}\)), that is \(\mathcal {I}_0^{(1)}(v_0)=c_0^{(1)}\). Then \(\mathcal {P}^{(1)}_{{\varepsilon }}(v_0)>0\) and there exists \(t_{v_0}({\varepsilon })>1\) such that \(v_0(x/t_{v_0}({\varepsilon }))\in \mathscr {P}^{(1)}_{{\varepsilon }}\). This implies that \(t_{v_0}({\varepsilon })\) is bounded, up to subsequence, we assume that \(t_{v_0}({\varepsilon })\rightarrow t_{v_0}(0)\). Recall that \(\mathcal {P}^{(1)}_0(v_0)=0\), we can conclude that \(t_{v_0}(0)=1\). Set \(\kappa :=\frac{q(2+\alpha )-2(2p+\alpha )}{4(p-1)}\). We obtain
which means that \(t^{\alpha }_{v_0}({\varepsilon })\le 1+C{\varepsilon }^{\kappa }\), or equivalently
Therefore
It follows that \(c_{{\varepsilon }}^{(1)}\rightarrow c_0^{(1)}\) as \({\varepsilon }\rightarrow 0\). \(\square \)
Corollary 7.2
The quantities
are uniformly bounded as \({\varepsilon }\rightarrow 0\).
Proof
Follows from Pohožaev and Nehari identities for \(\mathcal {I}_{{\varepsilon }}^{(1)}\) and Lemma 7.3, as in Corollary 7.1. \(\square \)
Proof of Theorem 2.5
Since
there exists \(t_{{\varepsilon }}\in (0,1)\) such that \(\mathcal {P}^{(1)}_{0}(v_{\varepsilon }(x/t_{{\varepsilon }}))=0\). Using Lemma 7.3, as \({\varepsilon }\rightarrow 0\) we obtain
Thus \(\{v_{\varepsilon }(x/t_{{\varepsilon }})\}\subset \mathscr {P}^{(1)}_{0}\) is a minimizing sequence for \(c_0^{(1)}\) and we may assume that \(t_{{\varepsilon }}\rightarrow t_0\in (0, 1]\). (If \(t_{{\varepsilon }}\rightarrow 0\) then \(0<c_0\le \mathcal {I}^{(1)}_0(v_{\varepsilon }(x/t_{\varepsilon }))\rightarrow 0\), which is a contradiction.) From Corollary 7.2 we see that \(\{v_{{\varepsilon }}(x/t_{{\varepsilon }})\}\) is bounded in \(H^1_{rad}\cap L^q({\mathbb {R}}^N)\). Then similarly to the argument in Step 3 of Sect. 4, using Strauss’ radial bounds, Lebesgue dominated convergence and nonlocal Brezis–Lieb Lemma [40, Proposition 4.7], we conclude that \(v_{\varepsilon }(x)\rightarrow \overline{v}_0\) in \(D^1\cap L^q({\mathbb {R}}^N)\) and \(\overline{v}_0\) is a groundstate solution of (\(\mathscr {C}_{\varepsilon }\)). \(\square \)
Remark 7.1
We claim that in fact \(t_0=1\). Indeed, if \(t_0<1\) then there exists \(\delta >0\) such that for all \({\varepsilon }\in (0, \delta )\), it holds \(t_{{\varepsilon }}<\frac{t_0+1}{2}\). Therefore,
as \({\varepsilon }\rightarrow 0\), which is a contradiction. In particular, since
we conclude that \(\varepsilon ^{\kappa }\Vert v_{\varepsilon }\Vert _q^q\rightarrow 0\) as \(\varepsilon \rightarrow 0\).
7.3 2nd rescaling: Thomas–Fermi limit for \(\alpha =2\)
Throughout this section we assume that \(N\le 5\), \(\alpha =2\), \(p=2\) and \(\frac{4N}{N+2}<q<3\). The energy corresponding to the 2nd rescaling
is given by
and the corresponding Pohožaev functional is
Note that \(\mathcal {I}_{\varepsilon }(u)={\varepsilon }^{\frac{q(N+2)-4N}{2(q-2)}}\mathcal {I}_{\varepsilon }^{(2)}(v)\) and consider the rescaled minimization problem
where \(\mathscr {P}_{{\varepsilon }}^{(2)}:=\left\{ u\ne 0: \mathcal {P}_{{\varepsilon }}^{(2)}(v)=0\right\} \) is the Pohožaev manifold of (\(T\!F_{\varepsilon }\)). When \({\varepsilon }=0\), we formally obtain
By using an appropriate rescaling, it is standard to see that the minimization problem for \(c_0^{(2)}\) is equivalent to the minimization problem \(s_{T\!F}\), defined in (2.3). In particular, \(c_0^{(2)}>0\) and \(c_0^{(2)}\) admits a minimizer \(v_0\in L^2\cap L^q({\mathbb {R}}^N)\), which is a groundstate of the Thomas–Fermi Eq. (\(T\!F\)) characterised in (2.6) of Theorem 2.6.
Let \(u_{\varepsilon }\) be the positive spherically symmetric groundstate solution of (\(P_{\varepsilon }\)) constructed in Theorem 1.1. It is clear that the rescaled groundstate
is a groundstate solution of (\(T\!F_{\varepsilon }\)), i.e. \(\mathcal {I}_{\varepsilon }^{(2)}(v_{\varepsilon })=c_{\varepsilon }^{(2)}\). We are going to show that \(v_{\varepsilon }\) converges to a groundstate of the Thomas–Fermi Eq. (\(T\!F\)), characterised in Theorem 2.6.
Before we do this, we deduce a two-sided estimate on the “boundary behaviour” of the the nonnegative radially symmetric Thomas–Fermi minimizer \(\rho _*\), constructed in Theorem 2.6. Recall that \(\mathrm {supp}(\rho _*)=\bar{B}_{R_*}\) for some \(R_*>0\) and \(\rho _*\) is \(C^\infty \) inside the support. Moreover, since we assume that \(p=2\), \(\alpha =2\) (and denote \(m=q/2\)), we see that \(\rho _*^{m-1}\in C^{0,1}({\mathbb {R}}^N)\) and \(\rho _*\in C^{0, \gamma }({\mathbb {R}}^N)\), where \(\gamma =\min \{1,\frac{1}{m-1}\}\). In particular, \(\rho _*\in H^1_0(B_{R_*})\) and we can apply \(-\Delta \) to the Euler–Lagrange Eq. (2.5) considered in \(B_{R_*}\), to obtain
We conclude that \(\rho _*^{m-1}\) is superharmonic in \(B_{R_*}\) and by the boundary Hopf lemma
Hence we deduce a two-sided bound
Similar estimates should be available for \(\alpha \ne 0\), at least under the assumption \(\alpha <2\). We will study this in the forthcoming paper [26].
Lemma 7.4
\(0<c^{(2)}_{{\varepsilon }}-c^{(2)}_0\rightarrow 0\) as \({\varepsilon }\rightarrow 0\).
Proof
Let \(v_{{\varepsilon }}\in \mathscr {P}^{(2)}_{{\varepsilon }}\) be a solution of \((P_{{\varepsilon }})\) with \(\mathcal {I}^{(2)}_{{\varepsilon }}(v_{{\varepsilon }})=c^{(2)}_{{\varepsilon }}\) and \({\varepsilon }>0\). Then
where we denoted \(\nu :=\frac{2(3-q)}{q-2}\). Let \(w_{{\varepsilon }, t}(x)=v_{{\varepsilon }}(\frac{x}{t})\), then we obtain that
has a unique maximum and \(\mathcal {P}_{0}^{(2)}(w_{{\varepsilon }, 1})<0\). Thus there exists \(t_{{\varepsilon }}\in (0,1)\) such that \(w_{{\varepsilon }, t_{{\varepsilon }}}\in \mathscr {P}_{0}^{(2)}\). Therefore we have
so \(c^{(2)}_0<c^{(2)}_{{\varepsilon }}\). We are going to prove that \(c^{(2)}_{{\varepsilon }}\rightarrow c^{(2)}_0\) as \({\varepsilon }\rightarrow 0\).
Let \(v_0\in \mathscr {P}^{(2)}_{0}\) be a groundstate (2.6) of the Thomas–Fermi Eq. (\(T\!F\)), as constructed in Theorem 2.6. From (2.6) and (7.17) we conclude that \(v_0^{q-2}\in C^{0,1}({\mathbb {R}}^N)\) and
where \(R_*\) is the support radius of \(v_0\) and
Note that if \(q\ge 4\) then \(v_0\not \in D^1({\mathbb {R}}^N)\) because of the singularity of the gradient on the boundary of the support, even if \(v_0^2\) is Lipschitz.
Given \(n\gg 1\), we introduce the cut-off function \(\eta _{n}\in C_c^{\infty }({\mathbb {R}}^N)\) such that \(\eta _n(x)=1\) for \(|x|\le R_*-\frac{1}{n}\), \(0<\eta _n(x)<1\) for \(R_*-\frac{1}{n}<|x|\le R_*-\frac{1}{2n}\), \(\eta _n(x)=0\) for \(|x|\ge R_*-\frac{1}{2n}\). Furthermore, \(|\eta _n'(x)|\le 4n\) and \(|\eta _n'(x)|\ge \frac{n}{2}\) for \(R_*-\frac{4}{5n}<|x|<R_*-\frac{3}{5n}\). Set
It is elementary to obtain the estimates
To estimate the gradient term, note that since \(v_0^{q-2}\) is Lipschitz on \({\mathbb {R}}^N\) and smooth inside the support, we have \(v_0^{q-3}|\nabla v_0|\in L^\infty ({\mathbb {R}}^N)\) and then it follows from (7.19) that
Then we have
On the other hand, using the right hand side of (7.19), we have
Therefore
Recall that \(N\le 5\), \(\alpha =2\), \(p=2\) and \(\frac{4N}{N+2}<q<3\) in this section, and hence \(\gamma _0=1/2\) (\(q\ge 3\) in (7.25) is needed to study the case \({\varepsilon }\rightarrow \infty \)). Then
Set \(n=\varepsilon ^{-\frac{3}{2}\nu }\). Then for \(\varepsilon >0\) small enough, we have
and there exists \(t_{\varepsilon }>1\) such that \(\mathcal {P}^{(2)}_{\varepsilon }(v_{\varepsilon }(x/t_{\varepsilon }))=0\). This implies that
It follows from (7.25) and (7.20) that \(t_{\varepsilon }\rightarrow 1\) as \(\varepsilon \rightarrow 0\). Moreover, we also have
Therefore, we have
which means that \(c^{(2)}_{\varepsilon }\rightarrow c_0^{(2)}\) as \(\varepsilon \rightarrow 0\). \(\square \)
Lemma 7.5
\({\varepsilon }^{\frac{2(3-q)}{q-2}}\Vert \nabla v_{{\varepsilon }}\Vert _2^2\rightarrow 0\) as \({\varepsilon }\rightarrow 0\).
Proof
According to Lemma 7.4, there exists a unique \(t_{{\varepsilon }}\in (0, 1)\) such that \(v_{{\varepsilon }}(x/t_{{\varepsilon }})\in \mathscr {P}^{(2)}_0\). Now we claim that \(t_{{\varepsilon }}\rightarrow 1\) as \({\varepsilon }\rightarrow 0\). If not, we assume that \(t_{{\varepsilon }}\rightarrow t_0<1\) as as \({\varepsilon }\rightarrow 0\), then by (7.1), we have, as \({\varepsilon }\rightarrow 0\),
this is a contradiction. Therefore our claim holds, i.e., \(t_{{\varepsilon }}\rightarrow 1\) as \({\varepsilon }\rightarrow 0\). Using \(\mathcal {P}^{(2)}_0(v_{{\varepsilon }}(x/t_{{\varepsilon }}))=0\) again, we see that
which implies that \({\varepsilon }^{\nu }\Vert \nabla v_{{\varepsilon }}\Vert _2^2\rightarrow 0\) as \({\varepsilon }\rightarrow 0\) since \(\mathcal {P}^{(2)}_{{\varepsilon }}(v_{{\varepsilon }})=0\). \(\square \)
Proof of Theorem 2.7
By Lemmas 7.4 and 7.5, we see that \(\{v_{{\varepsilon }}(x/t_{{\varepsilon }})\}\) is a bounded minimizing sequence of \(c^{(2)}_0\). Then similarly to the arguments in the proof of Proposition 6.1 and Theorem 2.6, there exists \(\overline{v}_0\in L^{2}({\mathbb {R}}^N)\cap L^{q}({\mathbb {R}}^N)\) such that
and \(\overline{v}_0\) is a weak solution of (\(T\!F\)).
8 Asymptotic profiles: critical regimes
8.1 Critical Choquard case
Throughout this sub-section we assume that \(p=\frac{N+\alpha }{N-2}\) and \(q>2^*=\frac{2N}{N-2}\). Consider the minimization problem
defined in (2.8). Combining Sobolev inequality (1.4) and HLS inequality (1.3),
hence \(\mathcal {S}_{HL}\ge \mathcal {S}_*\mathcal {C}_\alpha ^{-\frac{N-2}{N+\alpha }}\). It is not difficult to check (cf. [21, Lemma 1.1]) that \(\mathcal {S}_{HL}=\mathcal {S}_*\mathcal {C}_\alpha ^{-\frac{N-2}{N+\alpha }}\) and \(\mathcal {S}_{HL}\) is achieved by the function
and the family of rescalings
here \(U_*\) is the Emden–Fowler solution in (2.9). Up to a rescaling, \(V_\lambda \) is a solution of the critical Choquard Eq. (\(\mathscr {C}_{HL}\)) and satisfies
The energy functional associated to (\(\mathscr {C}_{HL}\)) is
We define
where
By a simple calculation, we see that \(c_{HL}=\frac{\alpha +2}{2(N+\alpha )}\mathcal {S}_{HL}^{\frac{N+\alpha }{\alpha +2}}\).
Lemma 8.1
Let \(\sigma _{{\varepsilon }}=c_{{\varepsilon }}-c_{HL}\). Then as \({\varepsilon }\rightarrow 0\), we have
Proof
Note that \(u_{{\varepsilon }}\in \mathscr {P}_{{\varepsilon }}\) is a solution of (\(P_{\varepsilon }\)) with \(\mathcal {I}_{{\varepsilon }}(u_{{\varepsilon }})=c_{{\varepsilon }}\), then \(\mathcal {P}_{HL}(u_{{\varepsilon }})<0\), thus there exists \(t_{{\varepsilon }}\in (0, 1)\) such that \(u_{{\varepsilon }}(x/t_{{\varepsilon }})\in \mathscr {P}_{HL}\) and we have
Therefore, \(\sigma _{{\varepsilon }}=c_{{\varepsilon }}-c_{HL}>0\).
Case \(N\ge 5\). Note that for \(N\ge 5\), \(V_{\lambda }(x)\in L^2({\mathbb {R}}^N)\) for each \(\lambda >0\), thus we see that \(\mathcal {P}_{{\varepsilon }}(V_{\lambda }(x))>0\). Then for each \({\varepsilon }>0\) and \(\lambda >0\), there exists a unique \(s_{{\varepsilon },\lambda }>1\) such that \(\mathcal {P}_{{\varepsilon }}(V(x/s_{{\varepsilon }, \lambda }))=0\), which means that
Then there exists \(\lambda _{{\varepsilon }}>0\) such that
This implies that \(s_{{\varepsilon }}:=s_{{\varepsilon }, \lambda _{{\varepsilon }}}\rightarrow 1\) as \({\varepsilon }\rightarrow 0\). Furthermore, we have
Therefore, we obtain that
which means that \(\sigma _{\varepsilon }\le C{\varepsilon }^{\frac{q(N-2)-2N}{(q-2)(N-2)}}\).
To consider the case \(N=3, 4\), given \(R\gg \lambda \), we introduce a cut-off function \(\eta _{R}\in C_c^{\infty }({\mathbb {R}}^N)\) such that \(\eta _R(r)=1\) for \(|r|<R\), \(0<\eta _R(r)<1\) for \(R<|r|<2R\), \(\eta _R(r)=0\) for \(|r|\ge 2R\) and \(|\eta _R'(r)|\le R/2\). Similarly, e.g. to [52, Section III, Theorem 2.1], we compute
Here we only sketch the proof of (8.4). We may assume that \(R/\lambda \gg 1\). Note that
It follows from [43, Lemma 6.2] that
Therefore, by (8.7), we estimate
Combining the above inequalities, we obtain (8.4).
When \(R\gg \lambda \), by the above estimates we conclude that \(\mathcal {P}_{{\varepsilon }}(\eta _RV_{\lambda })>0\). Then there exists a unique \(t_{{\varepsilon }}:=t_{{\varepsilon }}(R, \lambda )>1\) such that \(\mathcal {P}_{{\varepsilon }}(\eta _R(x/t_{{\varepsilon }})V_{\lambda }(x/t_{{\varepsilon }}))=0\), which implies
Case \(N=4\). For each \({\varepsilon }>0\) small, by (8.5), (8.6) and choosing
we have
Then it follows from (8.8) that \(t_{{\varepsilon }}\rightarrow 1\) as \({\varepsilon }\rightarrow 0\). Furthermore, we have \( t_{{\varepsilon }}\le 1+C\left( {\varepsilon }\ln \frac{1}{{\varepsilon }}\right) ^{\frac{q-4}{q-2}}. \)
Case \(N=3\). For each \({\varepsilon }>0\) small, by (8.5), (8.6) and choosing
we have
Then it follows from (8.8) that \(t_{{\varepsilon }}\rightarrow 1\) as \({\varepsilon }\rightarrow 0\). Furthermore, we have \( t_{{\varepsilon }}\le 1+C{\varepsilon }^{\frac{q-6}{2(q-4)}}. \)
Conclusion of the proof for \(N=3,4\). Combining previous estimates together, we obtain
which completes the proof of this lemma. \(\square \)
Lemma 8.2
Proof
Follows from Nehari and Pohožaev identities for (\(P_{\varepsilon }\)). \(\square \)
Lemma 8.3
\(\frac{(q-2)N{\varepsilon }}{q(N-2)-2N}\Vert u_{{\varepsilon }}\Vert _2^2\le C\sigma _{{\varepsilon }}\). Moreover, we have
Proof
By Lemma 8.1, we see that
This, together with Lemma 8.2, implies that there exists \(C>0\) such that
for all small \({\varepsilon }>0\). Note that \(\mathcal {P}_{HL}(u_{{\varepsilon }})<0\), then there exists \(t_{{\varepsilon }}\in (0, 1)\) such that \(\mathcal {P}_{HL}(u_{{\varepsilon }}(x/t_{{\varepsilon }}))=0\), which means that
If \(t_{{\varepsilon }}\rightarrow 0\) as \({\varepsilon }\rightarrow 0\), then we must have \(\Vert \nabla u_{{\varepsilon }}\Vert _2^2\rightarrow 0\) as \({\varepsilon }\rightarrow 0\), this contradicts
Therefore, there exists \(C>0\) such that \(t_{{\varepsilon }}>C\) for \({\varepsilon }>0\) small. Thus we have
This means that
Moreover, we have \({\varepsilon }\Vert u_{{\varepsilon }}\Vert _2^2\rightarrow 0\) as \({\varepsilon }\rightarrow 0\). It follows from Lemma 8.2 that
as \({\varepsilon }\rightarrow 0\).
Set \(w_{{\varepsilon }}(x)=u_{{\varepsilon }}(\mathcal {S}_{HL}^{\frac{1}{\alpha +2}}x)\), then we have, as \({\varepsilon }\rightarrow 0\),
On the other hand, by the HLS inequality, we have
This means that
which completes the proof. \(\square \)
Set
Then \(\Vert \overline{w}_{{\varepsilon }}\Vert _{2^*}=\Vert \overline{V}\Vert _{2^*}=1\), \(\Vert \nabla \overline{V}\Vert _2^2=\mathcal {S}_*\) and \(\{\overline{w}_{{\varepsilon }}\}\) is a minimizing sequence for the critical Sobolev constant \(\mathcal {S}_*\). Similarly to the arguments in [41, p. 1094], we conclude that for \({\varepsilon }>0\) small there exists \(\lambda _{{\varepsilon }}>0\) such that
We define the rescaled family
Then
i.e., \(\{v_{{\varepsilon }}\}\) is a minimizing sequence for \(\mathcal {S}_*\). Furthermore,
Lemma 8.4
\(\lim _{{\varepsilon }\rightarrow 0}\Vert \nabla (\overline{v}_{{\varepsilon }}-V)\Vert _2=\lim _{{\varepsilon }\rightarrow 0}\Vert \overline{v}_{{\varepsilon }}-V\Vert _{2^*}=0\), where \(\overline{v}_{{\varepsilon }}(x):=\lambda _{{\varepsilon }}^{\frac{N-2}{2}}u_{{\varepsilon }}(\lambda _{{\varepsilon }}x)\).
Proof
Since \(\{v_{{\varepsilon }}\}\) is a minimizing sequence of \(\mathcal {S}_*\), it follows from the Concentration-Compactness Principle of P.L.Lions [52, Chapter 1, Theorem 4.9] that
which, together with the definitions of \(w_{{\varepsilon }}\) and \(\overline{w}_{{\varepsilon }}\), implies that
\(\square \)
By a simple calculation, we see that \(v_{{\varepsilon }}\) solves the equation
By the definition of \(v_{{\varepsilon }}\) and \(w_{{\varepsilon }}\), we obtain
It follows from Lemma 8.2 and Lemma 8.3 that
Therefore we can deduce the following estimates on \(\lambda _{{\varepsilon }}\).
Lemma 8.5
\(\sigma _{{\varepsilon }}^{-\frac{2}{(N-2)q-2N}}\lesssim \lambda _{{\varepsilon }}\lesssim {\varepsilon }^{-\frac{1}{2}}\sigma _{{\varepsilon }}^{\frac{1}{2}}\), as \({\varepsilon }\rightarrow 0\).
Proof
Since \(v_{{\varepsilon }}\rightarrow \overline{V}\) strongly in \(D^{1}({\mathbb {R}}^N)\) and \(L^{2^*}({\mathbb {R}}^N)\), we have \(v_{{\varepsilon }}\rightarrow \overline{V}\) strongly in \(L^{s}_{loc}({\mathbb {R}}^N)\) for \(s\in [2, 2^*)\), thus we get that
and
Therefore, the assertion follows by (8.12), (8.13) and (8.14). \(\square \)
Set
By Lemma 8.5 and since \(u_{{\varepsilon }}\le C|x|^{-(N-2)/2}\Vert u_{{\varepsilon }}\Vert _{2^*}\), we conclude that
where \(C>0\) does not depend on \({\varepsilon }>0\) or \(x\in {\mathbb {R}}^N\). Therefore, for small \({\varepsilon }>0\), it follows from (8.10) that solutions \(v_{{\varepsilon }}>0\) satisfy the linear inequality
By [41, Lemma 4.8] we have the following lower estimates of \(v_{{\varepsilon }}\).
Proposition 8.1
There exists \(R>0\) and \(c>0\) such that for all small \({\varepsilon }>0\),
The proof of the next result is nearly identical to the proofs in [41, Lemma 4.6 and pp. 1097–1098]. We outline the arguments for reader’s convenience.
Lemma 8.6
For \({\varepsilon }\rightarrow 0\), we have \(\Vert v_{{\varepsilon }}\Vert _{q}^q\sim 1\),
Proof
For all \(N\ge 3\), (8.13) implies that \(\Vert v_{{\varepsilon }}\Vert _q^q\ge C\) for some \(C>0\).
Case \(N\ge 5\). By Lemmas 8.1 and 8.5, we see that for \(N\ge 5\)
Moreover, \(\Vert v_{{\varepsilon }}\Vert _2^2\sim 1\) and \(\Vert v_{{\varepsilon }}\Vert _q^q\sim 1\) by Lemma 8.1, (8.12), (8.13) and (8.14).
Case \(N=4\). By Proposition 8.1, we obtain
where \(\Gamma (0,t)=-\ln t-\gamma +\mathcal {O}(t)\) as \(t\searrow 0\) is the incomplete Gamma function and \(\gamma \approx 0.57\) is the Euler constant. Hence, by Lemmas 8.1 and 8.5, we obtain that
This, together with Lemmas 8.1, 8.5 and (8.12), implies that
Moreover,
Case \(N=3\). By Proposition 8.1, we obtain
This, together with Lemma 8.5 and (8.12), implies that
By Lemmas 8.1 and 8.5, we obtain that
Furthermore, we deduce from Lemma 8.1, (8.15) and (8.16) that
This complete the proof. \(\square \)
8.2 Critical Thomas–Fermi case
Throughout this section we assume that \(\frac{N+\alpha }{N}<p<\frac{N+\alpha }{N-2}\) and \(q=\frac{2Np}{N+\alpha }\). Let
where \(\mathcal {C}_{\alpha }\) is the optimal constant in (1.3), as described in (2.13).
It is well-known [34, Theorem 4.3] that \(\mathcal {S}_{T\!F}\) is achieved by the function
and the family of rescalings
here \(\widetilde{U}\) is the groundstate solution of (\(T\!F_*\)) defined in (2.14). It is clear that
The energy functional which corresponds to (\(T\!F_*\)) is
We define,
where
By a simple calculation, we see that \(c_{T\!F}=\frac{\alpha }{2Np}\mathcal {S}_{T\!F}^{\frac{N+\alpha }{\alpha }}\).
Note that \(\widetilde{V}_\lambda \in L^2({\mathbb {R}}^N)\) for \(N\ge 4\), and \(\widetilde{V}_\lambda \in D^{1}({\mathbb {R}}^N)\) if \(N\ge 3\).
Lemma 8.7
Let \(\overline{\sigma }_{{\varepsilon }}=c_{{\varepsilon }}-c_{T\!F}\) and \({\varepsilon }\rightarrow 0\). If \(N\ge 4\) then
while if \(N=3\) then
Proof
Note that \(u_{{\varepsilon }}\in \mathscr {P}_{{\varepsilon }}\) is a solution of (\(P_{\varepsilon }\)) with \(\mathcal {I}_{{\varepsilon }}(u_{{\varepsilon }})=c_{{\varepsilon }}\), then \(\mathcal {P}_{T\!F}(u_{{\varepsilon }})<0\), thus there exists \(t_{{\varepsilon }}\in (0, 1)\) such that \(u_{{\varepsilon }}(x/t_{{\varepsilon }})\in \mathscr {P}_{T\!F}\) and we have
Therefore, \(\overline{\sigma }_{{\varepsilon }}=c_{{\varepsilon }}-c_{T\!F}>0\).
Case \(N\ge 4\). Note that for \(N\ge 4\), \(\widetilde{V}_{\lambda }(x)\in L^2({\mathbb {R}}^N)\) for each \(\lambda >0\) and therefore, \(\mathcal {P}_{{\varepsilon }}(\widetilde{V}_{\lambda }(x))>0\). Then for each \({\varepsilon }>0\) and \(\lambda >0\), there exists a unique \(s_{{\varepsilon },\lambda }>1\) such that \(\mathcal {P}_{{\varepsilon }}(\widetilde{V}(x/s_{{\varepsilon }, \lambda }))=0\), which means that
then there exists \(\lambda _{{\varepsilon }}>0\) such that
Therefore \(s_{{\varepsilon }}:=s_{{\varepsilon }, \lambda _{{\varepsilon }}}\rightarrow 1\) as \({\varepsilon }\rightarrow 0\). Furthermore, we have
Therefore, we obtain that
which means that \(\sigma _{\varepsilon }\le C{\varepsilon }^{\frac{2N-q(N-2)}{2q}}\).
Case \(N=3\). To consider the case \(N=3\), given \(R\gg \lambda \), we introduce a cut-off function \(\eta _{R}\in C_c^{\infty }({\mathbb {R}}^N)\) such that \(\eta _R(r)=1\) for \(|r|<R\), \(0<\eta _R(r)<1\) for \(R<|r|<2R\), \(\eta _R(r)=0\) for \(|r|\ge 2R\) and \(|\eta _R'(r)|\le R/2\). We then compute as in, e.g. [52, Theorem 2.1]:
When \(R\gg \lambda \), by the above estimates, we see that \(\mathcal {P}_{{\varepsilon }}(\eta _R\widetilde{V}_{\lambda })>0\), then there exists a unique \(t_{{\varepsilon }}:=t_{{\varepsilon }}(R, \lambda )>1\) such that \(\mathcal {P}_{{\varepsilon }}(\eta _R(x/t_{{\varepsilon }})\widetilde{V}_{\lambda }(x/t_{{\varepsilon }}))=0\), which implies that
To estimate \(\psi _{{\varepsilon }}(R, \lambda )\) we consider three cases.
(i) Case \(p\in \left( \tfrac{1}{3}(3+\alpha ), \tfrac{2}{3}(3+\alpha )\right) \). For small \({\varepsilon }>0\), set
Then
Therefore it follows from (8.23) that \(t_{{\varepsilon }}\rightarrow 1\) as \({\varepsilon }\rightarrow 0\) and \( t_{{\varepsilon }}\le 1+C{\varepsilon }^{\frac{3+\alpha -p}{2p}}. \)
(ii) Case \(p=\tfrac{2}{3}(3+\alpha )\). For small \({\varepsilon }>0\), set
Then
Therefore it follows from (8.23) that \(t_{{\varepsilon }}\rightarrow 1\) as \({\varepsilon }\rightarrow 0\) and \( t_{{\varepsilon }}\le 1+C({\varepsilon }\ln \frac{1}{{\varepsilon }})^{\frac{1}{4}}. \)
(iii) Case \(p\in \left( \tfrac{2}{3}(3+\alpha ), 3+\alpha \right) \). For small \({\varepsilon }>0\), set
Then
Therefore it follows from (8.23) that \(t_{{\varepsilon }}\rightarrow 1\) as \({\varepsilon }\rightarrow 0\) and \( t_{{\varepsilon }}\le 1+C{\varepsilon }^{(\frac{3+\alpha -p}{p})^2}. \)
Conclusion of the proof for \(N=3\). From (i)–(iii) we deduce that
so the assertion follows. \(\square \)
Lemma 8.8
Proof
Follows from Nehari and Pohožaev identities for (\(P_{\varepsilon }\)). \(\square \)
Lemma 8.9
\(\frac{(q-2)N{\varepsilon }}{q(N-2)-2N}\Vert u_{{\varepsilon }}\Vert _2^2\le C\overline{\sigma }_{{\varepsilon }}\). Moreover, we have
Proof
By Lemma 8.7, we see that
This, together with Lemma 8.8, implies that there exists \(C>0\) such that \(\int _{{\mathbb {R}}^N}(I_{\alpha }*|u_{\varepsilon }|^{p})|u_{\varepsilon }|^{p}dx\ge C\) for all \({\varepsilon }\) small. Note that \(\mathcal {P}_{T\!F}(u_{{\varepsilon }})<0\), then there exists \(t_{{\varepsilon }}\in (0, 1)\) such that \(\mathcal {P}_{T\!F}(u_{{\varepsilon }}(x/t_{{\varepsilon }}))=0\), which means that
If \(t_{{\varepsilon }}\rightarrow 0\) as \({\varepsilon }\rightarrow 0\), then we must have \(\Vert u_{{\varepsilon }}\Vert _q^q\rightarrow 0\) as \({\varepsilon }\rightarrow 0\), this contradicts
Therefore, there exists \(C>0\) such that \(t_{{\varepsilon }}>C\) for \({\varepsilon }>0\) small. Then we have
this means that
Moreover, we have \({\varepsilon }\Vert u_{{\varepsilon }}\Vert _2^2\rightarrow 0\) as \({\varepsilon }\rightarrow 0\). It follows from Lemma 8.8 that
as \({\varepsilon }\rightarrow 0\). \(\square \)
Set \(w_{{\varepsilon }}(x)=u_{{\varepsilon }}(\mathcal {S}_{T\!F}^{\frac{1}{\alpha }}x)\). Then we have, as \({\varepsilon }\rightarrow 0\),
Let \(\overline{w}_{{\varepsilon }}(x)=w_{{\varepsilon }}(x)/\Vert w_{{\varepsilon }}(x)\Vert _q\) and \(\overline{V}(x):=\mathcal {S}_{T\!F}^{-\frac{1}{q}}\widetilde{V}(\mathcal {S}_{T\!F}^{\frac{1}{\alpha }}x)\), then we see that \(\Vert \overline{w}_{{\varepsilon }}\Vert _q^q=\Vert \overline{V}\Vert _q^q=1\) and
Thus \(\{\overline{w}_{{\varepsilon }}\}\) is an optimizing sequence for \(\mathcal {C}_\alpha \). Then similarly to the arguments in [41, Section 4.4] it follows that for \({\varepsilon }>0\) small there exists \(\lambda _{{\varepsilon }}>0\) such that
We define the rescaled family
then
i.e., \(\{v_{{\varepsilon }}\}\) is a maximizing sequence for \(\mathcal {C}_\alpha \). Moreover,
Lemma 8.10
\(\lim _{{\varepsilon }\rightarrow 0}\Vert \overline{v}_{{\varepsilon }}-\widetilde{V}\Vert _{q}=0\), where \(\overline{v}_{{\varepsilon }}(x):=\lambda _{{\varepsilon }}^{\frac{N+\alpha }{2p}}u_{{\varepsilon }}(\lambda _{{\varepsilon }}x)\).
Proof
It follows from Concentration-Compactness Principle of P.L.Lions [37, Theorem 2.1] that
which, together with the definitions of \(w_{{\varepsilon }}(x)\) and \(\overline{w}_{{\varepsilon }}(x)\), implies that
\(\square \)
By a simple calculation, we see that
solves the equation
By the definition of \(v_{{\varepsilon }}\) and \(w_{{\varepsilon }}\), we obtain
It follows from Lemma 8.8 and Lemma 8.9 that
Lemma 8.11
Let \({\varepsilon }\rightarrow 0\). If \(N\ge 4\) then
while if \(N=3\) then
Proof
Since \(v_{{\varepsilon }}\rightarrow \overline{V}\) strongly in \(L^{q}({\mathbb {R}}^N)\), we have \(v_{{\varepsilon }}\rightarrow \overline{V}\) strongly in \(L^{s}_{loc}({\mathbb {R}}^N)\) for \(s\in [2, q)\), thus we get that
and
Therefore, by (8.27), (8.30) and (8.31), we have
Then (8.28) and (8.29) follow directly from Lemma 8.7. \(\square \)
Lemma 8.12
If either \(N\ge 4\), or \(N=3\) and \(p\in \left( \frac{3+\alpha }{3}, \frac{2(3+\alpha )}{3}\right) \), then \(\Vert \nabla v_{{\varepsilon }}\Vert _2^2\sim 1\), \(\Vert v_{{\varepsilon }}\Vert _2^2\sim 1\).
Proof
Follows from (8.27), (8.30), (8.31) and Lemma 8.11. \(\square \)
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Acknowledgements
The authors are grateful to Augusto Ponce for communicating Theorem A.1, to Cyrill Muratov for helpful comments on the draft of the paper, and to Bruno Volzone for bringing to our attention [14] and for helpful comments. This work was initiated during a visit of Z.L. at Swansea University. Part of this work was conducted when V.M. visited Suzhou University of Science and Technology. The support and hospitality of both institutions are gratefully acknowledged.
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ZL was supported by NSFC Grant Numbers 11901418, 11771319, 12171470.
Appendix A: A contraction inequality
Appendix A: A contraction inequality
Consider the equation
where \(N\ge 2\) and \(q>2\). The existence for any \(f\in L^1_{loc}({\mathbb {R}}^N)\) of the unique distributional solution \(u_f\in L^1_{loc}({\mathbb {R}}^N)\) of (A.1) is the result in [9, Theorem 1]. The following remarkable contraction estimate on subsolutions for (A.1) was communicated to us by Augusto Ponce.
Theorem A.1
Let \(0\le f\in L^s({\mathbb {R}}^N)\) for some \(s\ge 1\) and let \(v\in L^1_{loc}({\mathbb {R}}^N)\) be a nonnegative distributional sub-solution of (A.1), i.e.
Then \(v^{q-1}\in L^s({\mathbb {R}}^N)\) and
The inequality (A.3) is an extension to unbounded domains of the result in [47, Proposition 4.24], see also [47, Exercise 4.15 and a hint on p. 402]. We outline the arguments for completeness.
Lemma A.1
Let \(0\le f\in L^s({\mathbb {R}}^N)\) for some \(s\ge 1\). For \(m\in {\mathbb {N}}\), let \(u_m\in L^1_{loc}(B_m)\) be the unique nonnegative distributional solution of
Then \(u_m\in L^s(B_m)\) and \(\Vert u_m^{q-1}\Vert _s\le \Vert f\Vert _s\).
Proof
Denote \(B_{m,\rho }:=B_m\cap \{|u_m|^{q-1}<\rho \}\). By the Cavalieri Principle (see e.g. [47, Proposition 1.7]), applied with the measurable function \(|u_m|^{q-1}\) and measure \(d\nu :=|u_m|^{q-1}_{\chi \{|u|^{q-1}<\rho \}}dx\), for every \(\rho >0\) we have
where in the last line we used the following key inequality proved in [47, (4.12) on p.67],
Applying once again Cavalieri’s Principle, this time with the measurable function \(|u_m|^{q-1}\) and measure \(d\bar{\nu }:=|f|_{\chi \{|u_m|^{q-1}<\rho \}}dx\), and using Hölder’s inequality, we have
which implies
Since the bound (A.5) holds uniformly for all \(m\in {\mathbb {N}}\) and \(\rho \rightarrow \infty \), the assertion follows. \(\square \)
Proof of Theorem A.1
Let \(m\in {\mathbb {N}}\) and \(\{u_m\}\) be the sequence of solutions to (A.4). Observe that \(u_{m+1}\) also solves the equation on \(B_m\) and \(u_{m+1}\ge 0\) on \(\partial B_m\). Thus, by the maximum principle on \(B_m\) one has \(u_{m+1}\ge u_m\), so \(\{u_m\}\) is an increasing sequence. Moreover, it follows from (A.5) that \(\{u_m\}\) is locally bounded in \(L^s({\mathbb {R}}^N)\). Then \(\{u_m\}\) converges pointwise to a function \(u_\infty \) that satisfies, by Fatou’s lemma,
By the monotone convergence theorem, \(u_m^{q-1}\rightarrow u_\infty ^{q-1}\) in \(L^s(B_m)\), for every \(m\in {\mathbb {N}}\). Hence \(u_\infty \) satisfies, for all \(\varphi \in C_c^{\infty }({\mathbb {R}}^N)\),
i.e. \(u_\infty \) is a distributional solution of (A.1). Then \(u_\infty =u_f\) is the unique solution of (A.1), by the Brezis’s uniqueness result [9, Theorem 1].
Now let \(v\in L^1_{loc}({\mathbb {R}}^N)\) be a nonnegative distributional sub-solution of (A.1) and set \(w=(v-u_f)^+\). By Kato’s inequality [9, Lemma A.1],
On the other hand, there is a \(\delta >0\) such that \(\delta (a-b)^{q-1}\le a^{q-1}-b^{q-1}\) for all \(a\ge b\ge 0\). Then
and \(w=0\) by [9, Lemma 2]. \(\square \)
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Liu, Z., Moroz, V. Limit profiles for singularly perturbed Choquard equations with local repulsion. Calc. Var. 61, 160 (2022). https://doi.org/10.1007/s00526-022-02255-y
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DOI: https://doi.org/10.1007/s00526-022-02255-y