Abstract
We consider a Dirichlet problem driven by the anisotropic (p(z), q(z))-Laplacian, with a parametric reaction exhibiting the combined effects of singular and concave-convex nonlinearities. The superlinear term may change sign. Using variational tools together with truncation and comparison techniques, we prove a global (for the parameter \(\lambda >0\)) existence and multiplicity theorem (a bifurcation-type theorem).
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1 Introduction
Let \(\Omega \subseteq {\mathbb {R}}^N\) be a bounded domain with a \(C^2\)-boundary \(\partial \Omega \). In this paper we study the following singular anisotropic Dirichlet problem
Given \(r \in C({\overline{\Omega }})\) with \(1<\min \limits _{{\overline{\Omega }}} r\), by \(\Delta _{r(z)}\) we denote the anisotropic r-Laplace differential operator defined by
In contrast to the isotropic r-Laplacian (that is, \(r(\cdot )\) is constant), the anisotropic operator is not homogeneous. In (\(P_\lambda \)) the equation is driven by the sum of two such operators with distinct variable exponents \(p(\cdot )\) and \(q(\cdot )\) (double phase problem). Given \(\vartheta \in L^\infty (\Omega )\), we set \(\vartheta _- = \text { ess}\inf \limits _{\Omega } \vartheta \) and \(\vartheta _+ = \text { ess}\sup \limits _{\Omega } \vartheta \). In (\(P_\lambda \)) we assume that \(1<\tau _-\le \tau _+< q_-\le q_+<p_-\le p_+\) and \(0<\eta _-\le \eta _+<1\). The perturbation f(z, x) is a Carathéodory function (that is, for all \(x \in {\mathbb {R}}\) \(z \rightarrow f(z,x)\) is measurable and for a.a. \(z \in \Omega \) \(x \rightarrow f(z,x)\) is continuous) which exhibits \((p_+-1)\)-superlinear growth as \(x \rightarrow +\infty \), but need not satisfy the usual in such cases Ambrosetti–Rabinowitz condition (the AR-condition for short) and may change sign (indefinite perturbation). So, problem (\(P_\lambda \)) in the reaction has the combined effects of singular and concave–convex nonlinearities with two distinguishing features. First the superlinear (convex) term need not satisfy the AR-condition and second this perturbation is in general sign-changing. In the past, anisotropic singular equations were studied without the presence of the concave term \(\lambda u^{\tau (z)-1}\) and with a superlinear perturbation which is positive. We refer to the works of Byun–Ko [2] and Saoudi–Ghanmi [21]. Both deal with equations driven by the anisotropic p-Laplacian only. More recently, Papageorgiou–Rădulescu–Zhang [19] considered singular anisotropic double phase problems with a superlinear positive perturbation and no concave term.
Closer to our work here is the recent paper of Papageorgiou–Winkert [14], who examined an isotropic version of problem (\(P_\lambda \)) (all the exponents of the problem are constant) with a superlinear positive perturbation. The definite sign of the perturbation allows the authors of [14] to produce an ordered pair of upper and lower solutions, which in turn leads to the nonemptiness of the set of admissible parameters. They prove a global existence and multiplicity result (a bifurcation-type theorem). Our aim in this paper is to extend their result to anisotropic problems with an indefinite superlinear perturbation.
Finally we mention also the recent works on some other classes of anisotropic singular problems of Papageorgiou–Winkert [13, 15] and Papageorgiou–Zhang [16, 17]; for problems in divergence form, some recent results are given in Abdalmonem–Scapellato [1], Ragusa [20] and Wei [23] for parabolic equations.
2 Mathematical Background: Hypotheses
The analysis of problem (\(P_\lambda \)) is based on the variable Lebesgue and Sobolev spaces. A comprehensive introduction to the subject can be found in the books of Cruz Uribe–Fiorenza [3] and of Diening–Harjulehto–Hästö–Ru̇žička [4].
We introduce the set
Recall that for \(r \in C({\overline{\Omega }})\), \(r_-=\min \limits _{{\overline{\Omega }}}r\) and \(r_+=\max \limits _{{\overline{\Omega }}}r.\) Let \(L^0(\Omega )\) be the space of all measurable functions \(u:\Omega \rightarrow {\mathbb {R}}\). As usual we identify two such functions which differ only on a Lebesgue null subset of \(\Omega \). Given \(r \in E_1\), the variable Lebesgue space \(L^{r(z)}(\Omega )\) is defined by
We endow this space with the so-called “Luxemburg norm” defined by
With this norm the space \(L^{r(z)}(\Omega )\) becomes a separable and uniformly convex (thus reflexive, see [12], p. 225) Banach space. Let \(r^\prime \in E_1\) be the conjugate variable exponent to \(r(\cdot )\), defined by
We have that
and the following Hölder-type inequality holds
If \(r_1,r_2 \in E_1\) and \(r_1(z) \le r_2(z)\) for all \(z \in {\overline{\Omega }}\), then
Using the variable Lebesgue spaces, we can define the corresponding variable Sobolev spaces. So, given \(r \in E_1\), the variable Sobolev space \(W^{1,r(z)}(\Omega )\) is defined by
with \(\nabla u\) being the weak gradient of \(u(\cdot )\). This space is equipped with the following norm
with \( \Vert \nabla u\Vert _{r(z)} = \Vert |\nabla u | \Vert _{r(z)} \).
By \(C^{0,1}({\overline{\Omega }})\) we denote the space of all Lipschitz continuous functions \(u: {\overline{\Omega }} \rightarrow {\mathbb {R}}\). Given \(r \in C^{0,1}({\overline{\Omega }}) \cap E_1 \), we define
Both spaces \( W^{1,r(z)}(\Omega )\) and \( W_0^{1,r(z)}(\Omega )\) are separable and uniformly convex (thus reflexive) Banach spaces. Since in the definition of \(W_0^{1,r(z)}(\Omega )\) we assume that the exponent \(r(\cdot )\) is Lipschitz continuous, the Poincaré inequality holds, that is, there exists \(c=c(\Omega )>0\) such that
The Poincaré inequality leads to the following equivalent norm on \(W_0^{1,r(z)}(\Omega )\)
In the sequel we will use this norm on \(W_0^{1,r(z)}(\Omega )\). For \(r \in E_1\), we introduce the corresponding critical Sobolev exponent \(r^*(\cdot )\) given by
There is an anisotropic version of the Sobolev embedding theorem.
Proposition 1
If \(r \in C^{0,1}({\overline{\Omega }}) \cap E_1 \), \(r_+<N\), \(q \in E_1 \) and \(q(z) \le r^*(z)\) (resp. \(q(z) < r^*(z)\)) for all \(z \in {\overline{\Omega }}\), then \( W_0^{1,r(z)}(\Omega ) \hookrightarrow L^{q(z)}(\Omega )\) continuously (resp. compactly).
There is a close relation between the norm \(\Vert \cdot \Vert _{r(z)}\) and the modular function \(\rho _r(u)=\int _\Omega |u|^{r(z)}dz\).
Proposition 2
Suppose \(r \in E_1\) and \(\{u_n,u \}_{n \in {\mathbb {N}}} \subseteq L^{r(z)}(\Omega )\), then we have:
-
(a)
\(\Vert u\Vert _{r(z)}=\lambda \, \Leftrightarrow \, \rho _r\left( \frac{u}{\lambda }\right) =1\) (\(\lambda >0\)).
-
(b)
\(\Vert u\Vert _{r(z)}<1 \, (\text { resp. } =1,>1) \Leftrightarrow \, \rho _r(u)<1 (\text { resp. } =1, >1)\).
-
(c)
\(\Vert u\Vert _{r(z)}<1 \, \Rightarrow \, \Vert u\Vert _{r(z)}^{r_+}\le \rho _r(u) \le \Vert u\Vert _{r(z)}^{r_-}\).
-
(d)
\(\Vert u\Vert _{r(z)}>1 \Rightarrow \Vert u\Vert _{r(z)}^{r_-}\le \rho _r(u) \le \Vert u\Vert _{r(z)}^{r_+}\).
-
(e)
\(\Vert u_n\Vert _{r(z)}\rightarrow 0 \, (\text { resp. } \rightarrow +\infty ) \, \Leftrightarrow \, \rho _r(u_n)\rightarrow 0 \, (\text { resp. } \rightarrow +\infty )\).
We know that
Let \(A_{r}: W_0^{1,r(z)}(\Omega ) \rightarrow W^{-1,r^\prime (z)}(\Omega )\) be defined by
This operator has the following properties (see Gasiński–Papageorgiou [6], Proposition 2.5).
Proposition 3
The operator \(A_{r}(\cdot )\) is bounded (that is, maps bounded sets to bounded sets), continuous, strictly monotone (thus maximal monotone too) and of type \((S)_+\) (that is, if \(u_n \xrightarrow {w} u \) in \(W_0^{1,r(z)}(\Omega )\) and \(\limsup \limits _{n \rightarrow +\infty } \langle A_{r}(u_n), u_n - u \rangle \le 0\), then \(u_n \rightarrow u\) in \(W_0^{1,r(z)}(\Omega )).\)
The anisotropic regularity theory (see Fan [5] and Lieberman [11] for the corresponding isotropic theory) will lead us to the space \(C_0^1({\overline{\Omega }})=\{u \in C^1({\overline{\Omega }}): u \big |_{\partial \Omega }=0\}\). This is an ordered Banach space with positive (order) cone \(C_+=\left\{ u \in C_0^1({\overline{\Omega }}): u(z) \ge 0\right. \) \(\left. \text { for all } z \in {\overline{\Omega }} \right\} \). This cone has a nonempty interior given by
where \(\dfrac{\partial u}{\partial n}=(\nabla u, n)_{{\mathbb {R}}^N}\) with \(n(\cdot )\) being the outward unit normal on \(\partial \Omega \).
Let \(u_1,u_2 \in L^0( \Omega )\) such that \(u_1(z) \le u_2(z)\) for a.a. \(z \in \Omega \). We introduce the following sets:
If \(h_1,h_2 \in L^0( \Omega )\), then we say that \(h_1 \prec h_2\) if and only if for every \(K \subseteq \Omega \) compact we have
Evidently, if \(h_1,h_2 \in C(\Omega )\) and \(h_1(z) <h_2(z)\) for all \(z \in \Omega \), then \(h_1 \prec h_2\).
Given \(h \in L^0(\Omega )\), we set
We have \(h^\pm \in L^0(\Omega )\), \(h=h^+-h^-\), \( |h|=h^++h^-\) and if \(h \in W_0^{1,p(z)}(\Omega )\), then \(h^\pm \in W_0^{1,p(z)}(\Omega )\).
Let X be a Banach space and \(\varphi \in C^1(X,{\mathbb {R}})\). By \( K_\varphi \) we denote the critical set of \(\varphi (\cdot )\), that is
We say that \(\varphi (\cdot )\) satisfies the “C-condition”, if it has the following property:
“Every sequence \(\{u_n\}_{n \in {\mathbb {N}}} \subseteq X\) such that
\(\{\varphi (u_n)\}_{n \in {\mathbb {N}}} \subseteq {\mathbb {R}}\) is bounded
and \((1 + \Vert u_n\Vert _X) \varphi '(u_n) \rightarrow 0\) in \(X^*\) as \(n \rightarrow +\infty \),
admits a strongly convergent subsequence”.
The hypotheses on the data of (\(P_\lambda \)) are the following:
- \(H_0\):
-
: \(p,q \in C^{0,1}({\overline{\Omega }})\), \(\tau \in C({\overline{\Omega }})\), \(1<\tau _- \le \tau _+<q_-\le q_+<p_-\le p_+<N\), \(\eta \in C({\overline{\Omega }})\), \(0<\eta _-\le \eta _+<1\).
- \(H_1\):
-
: \(f:\Omega \times {\mathbb {R}} \rightarrow {\mathbb {R}}\) is a Carathéodory function such that \(f(z,0)=0\) for a.a. \(z \in \Omega \), and
- (i):
-
\(|f (z,x)| \le a(z) [1+x^{r(z)-1}]\) for a.a. \(z \in \Omega \), all \(x \ge 0\), with \(a \in L^\infty (\Omega )_+\), \(r \in C({\overline{\Omega }})\), \(p_+< r_-\le r_+ < p_-^*\);
- (ii):
-
if \(F(z,x)=\int _0^x f(z,s)\mathrm{{d}}s\), then \(\lim \limits _{x \rightarrow +\infty } \frac{F(z,x)}{x^{p_+}}=+\infty \) uniformly for a.a. \(z \in \Omega \);
- (iii):
-
if \(e_\lambda (z,x)=\lambda \left[ x^{1-\eta (z)}+x^{\tau (z)}\right] +f(z,x)x-\lambda p_+\left[ \frac{1}{1-\eta (z)}x^{1-\eta (z)}+ \frac{1}{\tau (z)}x^{\tau (z)}\right] +p_+ F(z,x)\), then there exists \(\vartheta \in L^1(\Omega )\) such that \(e_\lambda (z,x)\le e_\lambda (z,y)+\vartheta (z)\) for a.a. \(z\in \Omega \), all \(0\le x\le y\);
- (iv):
-
\( \lim \limits _{x \rightarrow 0^+} \frac{f(z,x)}{x^{q_+-1}}=0\) uniformly for a.a. \(z \in \Omega \), there exists \(\delta >0\) such that \(0<m_s \le f(z,x)\) for a.a. \(z \in \Omega \), all \(0<s\le x\le \delta \), and for every \(\rho >0\) there exists \({\widehat{\xi }}_\rho >0\) such that for a.a. \(z \in \Omega \), the function \(x \rightarrow f(z,x) + {\widehat{\xi }}_\rho |x|^{p(z)-1}\) is nondecreasing on \([0,\rho ]\).
Remark 1
Since we are looking for positive solutions and the above hypotheses concern the positive semiaxis \({\mathbb {R}}_+=[0,+\infty )\), without any loss of generality we may assume that \(f(z,x)=0\) for a.a. \(z \in \Omega \), all \(x\le 0\). Hypotheses \(H_1\) (ii), (iii) imply that
So, \(f(z,\cdot )\) is \((p_+-1)\)-superlinear, but need not satisfy the AR-condition which is common in the literature when studying superlinear problems (see Willem [24], p. 46). Instead we use the quasimonotonicity condition on \(e_\lambda (z,\cdot )\) (see hypothesis \(H_1\) (iii)). This is a slight generalization of a condition used by Li-Yang [10]. If there exists \(M>0\) such that for a.a. \(z \in \Omega \), \(x\rightarrow \frac{f(z,x)}{x^{p_+-1}}\) is nondecreasing on \([M,+\infty )\), then hypothesis \(H_1\) (iii) is satisfied. We stress that in contrast to [14], the perturbation here can be sign-changing.
Let \(V: W_0^{1,p(z)}(\Omega ) \rightarrow W^{-1,p^\prime (z)}(\Omega )\) be defined by
Evidently \(V=A_p+A_q\) and so on account of Proposition 3, we have:
Proposition 4
The operator \(V(\cdot )\) is bounded, continuous, strictly monotone (thus maximal monotone too) and of type \((S)_+\).
3 An Auxiliary Problem
In this section, we examine the following auxiliary anisotropic Dirichlet problem
The solution of this problem will help us bypass the singularity and prove the existence of admissible parameters for problem (\(P_\lambda \)).
Proposition 5
If hypothesis \(H_0\) holds, then for every \(\lambda >0\) problem (\(Q_\lambda \)) has a unique positive solution \({\overline{u}}_\lambda \in \mathrm{int \,}C_+\) and \({\overline{u}}_\lambda \rightarrow 0\) in \(C_0^1({\overline{\Omega }})\) as \(\lambda \rightarrow 0^+\).
Proof
First we show the existence of a positive solution for problem (\(Q_\lambda \)). To this end let \(\sigma _\lambda : W_0^{1,p(z)}(\Omega ) \rightarrow {\mathbb {R}}\) be the \(C^1\)-functional defined by
for all \(u \in W^{1,p(z)}_0(\Omega )\). If \(\Vert u\Vert ,\Vert u\Vert _{\tau (z)} \ge 1\), then we have
Since \(\tau _+<q_-<p_-\), it follows that
\(\sigma _\lambda (\cdot )\) is coercive.
The modular functions are convex continuous, hence sequentially weakly lower semi-continuous. This fact and Proposition 1 (the anisotropic Sobolev embedding theorem) imply that
\(\sigma _\lambda (\cdot )\) is sequentially weakly lower semicontinuous.
Then the Weierstrass–Tonelli theorem implies that there exists \({\overline{u}}_\lambda \in W^{1,p(z)}_0(\Omega )\) such that
Let \(u \in W^{1,p(z)}_0(\Omega )\), \(u \ne 0\). Then for \(t\in (0,1)\) we have
Since \(\tau _+<q_-\), choosing \(t\in (0,1)\) even smaller if necessary, we have
From (1) we have
In (2) we use the test function \(h=-{\overline{u}}_\lambda ^- \in W_0^{1,p(z)}(\Omega )\) and obtain
From (2) and (3) it follows that \({\overline{u}}_\lambda \) is a positive solution of (\(Q_\lambda \)). From [19] (Proposition A1), we have that \({\overline{u}}_\lambda \in L^\infty (\Omega )\). Then the anisotropic regularity theory (see Fan [5]) implies that \({\overline{u}}_\lambda \in C_+ \setminus \{0\}\). Finally the anisotropic maximum principle (see [19], Proposition A2) implies that
Next we show that this positive solution of (\(Q_\lambda \)) is unique. For \(\tau _0 \in (\tau _+,q_-)\), we consider the integral functional \(j: L^1(\Omega )\rightarrow \overline{{\mathbb {R}}}={\mathbb {R}} \cup \{+\infty \}\) defined by
Theorem 2.2 of Takác̆–Giacomoni [22] implies that \(j(\cdot )\) is convex. Suppose \({\widetilde{u}}_\lambda \) is another positive solution of (\(Q_\lambda \)). Again we have \({\widetilde{u}}_\lambda \in \mathrm{int \, }C_+\). Using Proposition 4.1.22, p. 274, of Papageorgiou–Rădulescu–Repovš [18], we have
Let \(\mathrm{dom \,}j=\{u \in L^1(\Omega ): j(u)< +\infty \}\) (the effective domain of \(j(\cdot )\)) and let \(h= ({\overline{u}}_\lambda ^{\tau _0}-{\widetilde{u}}_\lambda ^{\tau _0}) \in W_0^{1,p(z)}(\Omega )\). On account of (4) for \(t\in (0,1)\) small we have
Then since \(j(\cdot )\) is convex, the directional derivatives of \(j(\cdot )\) at \({\overline{u}}_\lambda ^{\tau _0}\) and at \({\widetilde{u}}_\lambda ^{\tau _0}\) in the direction h exist and using Green’s identity we have
The convexity of \(j(\cdot )\) implies the monotonicity of \(j^\prime (\cdot )\). Hence
This proves the uniqueness of the positive solution \({\overline{u}}_\lambda \in \mathrm{int \,}C_+\) of (\(Q_\lambda \)).
Finally we have
Using \(h={\overline{u}}_\lambda \in W_0^{1,p(z)}(\Omega )\), we obtain
Recall that \(\tau _+<q_-<p_-\). So, it follows that
The anisotropic regularity theory (see Fan [5]) implies that we can find \(\alpha \in (0,1)\) and \(c_4>0\) such that
We know that \(C_0^{1,\alpha }({\overline{\Omega }}) \hookrightarrow C_0^{1}({\overline{\Omega }})\) compactly. So, from (6) and (5) we conclude that
\(\square \)
4 Positive Solutions
We introduce the following two sets:
Proposition 6
If hypotheses \(H_0\), \(H_1\) hold, then \({\mathcal {L}} \ne \emptyset \) and for all \(\lambda >0\) \(S_\lambda \subseteq \mathrm{int \,}C_+\).
Proof
Let \(\delta >0\) be as postulated by hypothesis \(H_1 (iv)\). On account of Proposition 5, we can find \(\lambda ^*>0\) such that
We fix \(\lambda \in (0,\lambda ^*]\) and let \({\overline{u}}_\lambda \in \mathrm{int \,}C_+\) be the unique positive solution of (\(Q_\lambda \)) (see Proposition 5). We introduce the Carathéodory function \(g_\lambda (z,x)\) defined by
Let \(G_\lambda (z,x)=\int _0^x g_\lambda (z,s)ds\) and consider the \(C^1\)-functional \(\varphi _\lambda :W_0^{1,p(z)}(\Omega )\rightarrow {\mathbb {R}}\) defined by
Claim: \(\varphi _\lambda (\cdot )\) satisfies the C-condition.
Consider a sequence \(\{u_n\}_{n \in {\mathbb {N}}}\subseteq W_0^{1,p(z)}(\Omega )\) such that
From (10) we have
In (11) we choose the test function \(h=-u_n^- \in W_0^{1,p(z)}(\Omega )\). Using (8) we obtain
We define
and \({\widehat{F}}_\lambda (z,x)=\int _0^x {\widehat{f}}_\lambda (z,s)\mathrm{{d}}s\).
From (12), (9) and (8), we have
In (11) we choose the test function \(h=u_n^+ \in W_0^{1,p(z)}(\Omega )\) and obtain
We add (13) and (14) and obtain
(note that \(e_\lambda (z,x)={\widehat{f}}_\lambda (z,x)x-p_+{\widehat{F}}_\lambda (z,x)\) for all \(z \in \Omega \), all \(x \ge 0\)). Using (15) we will show that \(\{u_n^+\}_{n \in {\mathbb {N}}}\subseteq W_0^{1,p(z)}(\Omega )\). Arguing by contradiction, suppose that at least for a subsequence we have
We set \(y_n = \dfrac{u_n^+}{\Vert u_{n}^+ \Vert }\), \(n \in {\mathbb {N}}\). Then \(y_n \in W_0^{1,p(z)}(\Omega )\), \(y_n \ge 0\), \(\Vert y_n \Vert = 1\) for all \(n \in {\mathbb {N}}\). So, we may assume that
Suppose \(y \ne 0\). We set \(\Omega _+=\{z\in \Omega : y(z)>0\}\). From (17) we see that \(|\Omega _+|_N>0\) (by \(|\cdot |_N\) we denote the Lebesgue measure on \({\mathbb {R}}^N\)). We have
Then from (18), hypothesis \(H_1 (ii)\) and since \(\tau _+<p_+\), we see that
Using Fatou’s lemma, we have
From (12), (9) and (8), we have
Comparing (20) and (19), we have a contradiction.
Next suppose that \(y=0\). Consider the function
The function \(\mu _n(\cdot )\) is continuous and we can find \(t _n\in [0,1]\) such that
Let \(\beta >1\) and set \(v_n=(2\beta )^{1/p_-}y_n\), \(n \in {\mathbb {N}}\). From (17) and since we assume that \(y=0\), we have
From (16) we see that we can find \(n_0\in {\mathbb {N}}\) such that
Then from (21) and (23) we see that
From (22) we see that there exists \(n_1 \in {\mathbb {N}}\), \(n_1\ge n_0\), such that
But \(\beta >1\) is arbitrary. So, we infer that
We have
We set
Then from (8), (7) and hypothesis \(H_1 (iv)\), we see that
Note that
From (24) and (27) it follows that there exists \(n_2 \in {\mathbb {N}}\) such that
Then (28) and (21) imply that for all \(n \ge n_2\), we have
For \(n \ge n_2\) we have
We compare (24) and (30) and reach a contradiction. Therefore \(\{u_n^+\}_{n \in {\mathbb {N}}}\subseteq W_0^{1,p(z)}(\Omega )\) is bounded and this combined with (12) implies that \(\{u_n\}_{n \in {\mathbb {N}}}\subseteq W_0^{1,p(z)}(\Omega )\) is bounded. We may assume that
In (11) we use the test function \(h=(u_n -u) \in W_0^{1,p(z)}(\Omega )\), pass to the limit as \(n \rightarrow + \infty \) and use (31). We obtain
This proves the Claim.
On account of hypotheses \(H_1 (i),(iv)\), given \(\varepsilon >0\), we can find \(c_{19}=c_{19}(\varepsilon )>0\) such that
Consider \(u \in W_0^{1,p(z)}(\Omega )\) with \(\Vert u\Vert \le 1\) small. We have
Let \({\widehat{d}}(z)=d(z,\partial \Omega )\) for all \(z \in {\overline{\Omega }}\). Then Lemma 14.16, p. 355, of Gilbarg–Trudinger [8] implies that \({\widehat{d}} \in C_+ \setminus \{0\}\). Since \({\overline{u}}_\lambda \in \textrm{int }\, C_+\) (see Proposition 5), using Proposition 4.1.22, p. 274, of [18], we can find \(c_{20}>0\) such that
Using the anisotropic Hardy’s inequality of Harjulehto–Hästo–Koskenoja [9], we have
Also we have
and
We return to (33) and use (35), (36), (37) and (32). We obtain
Choosing \(\varepsilon >0\) small, we have
Consider the function
Evidently \({\widehat{\gamma }}_\lambda \in C^1(0,\infty )\) and
Therefore we can find \(t_0 \in (0,1)\) such that
We have
Since \(p_+<r_-\), we see that
Therefore we can find \({\widehat{\lambda }}_0>0\) such that
Let \({\overline{B}}_\lambda = \{u\in W_0^{1,p(z)}(\Omega ): \Vert u\Vert \le t_0(\lambda ) \}\). The reflexivity of \(W_0^{1,p(z)}(\Omega )\) and the Eberlein–Šmulian theorem imply that \({\overline{B}}_\lambda \) is sequentially weakly compact. Also, the sequential weak lower semicontinuity of the modular function and Proposition 1 imply that \(\varphi _\lambda (\cdot )\) is sequentially weakly lower semicontinuous. So, we can find \(u_\lambda \in {\overline{B}}_\lambda \) such that
Recall that \({\overline{u}}_\lambda \in \mathrm{int \,}C_+\). So, if \(u \in C_+ {\setminus } \{0\}\), we can find \(t \in (0,1)\) small such that
Using (8) and hypothesis \(H_1 (iv)\), we have
Since \(1<q_-\), choosing \(t\in (0,1)\) even smaller if necessary, we have
Then from (38) we see that
We use the test function \(h=({\overline{u}}_\lambda -u_\lambda )^+ \in W_0^{1,p(z)}(\Omega )\). We have
From (41), (8) and (40), we infer that
\(u_\lambda \) is a positive solution of (\(P_\lambda \)).
From Proposition A1 of Papageorgiou–Rădulescu–Zhang [19], we know that \(u_\lambda \in L^\infty (\Omega )\). Then the singular anisotropic regularity theory (see Saoudi–Ghanmi [21] and Giacomoni–Kumar–Sreenadh [7] for the corresponding isotropic theory) implies that \(u_\lambda \in C_+{\setminus } \{0\}\). Let \(\rho =\Vert u_\lambda \Vert _\infty \) and let \({\widehat{\xi }}_\rho >0\) be as postulated by hypothesis \(H_1 (iv)\). We have
We conclude that
\({\mathcal {L}} \ne \emptyset \) and \(S_\lambda \subseteq \mathrm{int \,}C_+\) for all \(\lambda >0\).
\(\square \)
The next proposition establishes a structural property of the set \({\mathcal {L}}\), namely that it is connected.
Proposition 7
If hypotheses \(H_0\), \(H_1\) hold, \(\lambda \in {\mathcal {L}}\) and \(0<\mu <\lambda \), then \(\mu \in {\mathcal {L}}\).
Proof
Let \(u_\lambda \in S_\lambda \subseteq \mathrm{int \,}C_+\). From Proposition 5 we know that \({\overline{u}}_\sigma \rightarrow 0\) in \(C_0^1({\overline{\Omega }})\) as \( \sigma \rightarrow 0^+\). So, we can find \(\sigma \in (0,\mu )\) small such that
We introduce the Carathéodory function \({\widehat{g}}_\mu (z,x)\) defined by
We set \({\widehat{G}}_\mu (z,x)=\int _0^x {\widehat{g}}_\mu (z,s)ds\) and consider the \(C^1\)-functional \({\widehat{\psi }}_\mu :W_0^{1,p(z)}(\Omega )\rightarrow {\mathbb {R}}\) defined by
From (42) and Proposition 2, it is clear that \({\widehat{\psi }}_\mu (\cdot )\) is coercive. Also, it is sequentially weakly lower semicontinuous (see Proposition 1). Then by the Weierstrass-Tonelli theorem, we can find \(u_\mu \in W_0^{1,p(z)}(\Omega )\) such that
In (43) first we use the test function \(h=({\overline{u}}_\sigma -u_\mu )^+ \in W^{1,p(z)}_0(\Omega )\). We have
Next in (43) we choose the test function \(h=(u_\mu -u_\lambda )^+ \in W^{1,p(z)}_0(\Omega )\). We have
So, we have proved that
Then from (44), (42) and (43) it follows that
\(u_\mu \in S_\mu \subseteq \mathrm{int \,}C_+\) and so \(\mu \in {\mathcal {L}}\).
\(\square \)
A quick inspection of the above proof reveals that we get, as a useful byproduct of it, the following corollary.
Corollary 1
If hypotheses \(H_0\), \(H_1\) hold, \(\lambda \in {\mathcal {L}}\), \(u_\lambda \in S_\lambda \subseteq \mathrm{int \,}C_+\) and \(\mu \in (0,\lambda )\), then \(\mu \in {\mathcal {L}}\) and there exists \(u_\mu \in S_\mu \subseteq \mathrm{int \,}C_+\) such that \(u_\mu \le u_\lambda \).
In fact with little additional effort, we can improve the above “monotonicity” property of the solution multifunction \(\lambda \rightarrow S_\lambda \).
Proposition 8
If hypotheses \(H_0\), \(H_1\) hold, \(\lambda \in {\mathcal {L}}\), \(u_\lambda \in S_\lambda \subseteq \mathrm{int \,}C_+\) and \(\mu \in (0,\lambda )\), then \(\mu \in {\mathcal {L}}\) and there exists \(u_\mu \in S_\mu \subseteq \mathrm{int \,}C_+\) such that \(u_\lambda -u_\mu \in \mathrm{int \,}C_+\).
Proof
From Corollary 1 we already know that \(\mu \in {\mathcal {L}}\) and there exists \(u_\mu \in S_\mu \subseteq \mathrm{int \,}C_+\) such that
Let \(\rho =\Vert u_\lambda \Vert _\infty \) and let \({\widehat{\xi }}_\rho >0\) be as postulated by hypothesis \(H_1(iv)\). We have
Since \(u_\mu \in \mathrm{int \, } C_+\), we see that \(0 \prec (\lambda -\mu )u_\mu ^{\tau (z)-1}\). Hence from (45) and Proposition 2.3 of Papageorgiou-Winkert [13], we obtain
\(\square \)
From the proof of Proposition 7, we know that for \(\sigma \in (0,\mu )\) small, we have
In fact using Proposition 8, we can improve this.
Proposition 9
If hypotheses \(H_0\), \(H_1\) hold, \(\lambda \in {\mathcal {L}}\), \(u_\lambda \in S_\lambda \subseteq \mathrm{int \,}C_+\) and \(\mu \in (0,\lambda )\), then we can find \(u_\mu \in S_\mu \subseteq \mathrm{int \,}C_+\) and \(\sigma \in (0,\mu )\) small such that \(u_\mu \in \textrm{int}_{C_0^1({\overline{\Omega }})}[{\overline{u}}_\sigma ,u_\lambda ]\).
Proof
From Proposition 8, we already know that there exists \(u_\mu \in S_\mu \subseteq \mathrm{int \,}C_+\) such that
Also if \(\sigma \in (0,\mu )\) is small, we have \({\overline{u}}_\sigma \le \min \{\delta ,u_\mu \}\) (see Proposition 5). Let \(\rho =\Vert u_\mu \Vert _\infty \) and let \({\widehat{\xi }}_\rho >0\) be as postulated by hypothesis \(H_1(iv)\). We have
Since \({\overline{u}}_\sigma \in \mathrm{int \, } C_+\), on account of hypothesis \(H_1(iv)\), we have
So, from (47) and Proposition 2.3 of Papageorgiou-Winkert [13], we infer that
\(\square \)
Let \({\widehat{\lambda }}=\sup {\mathcal {L}}.\)
Proposition 10
If hypotheses \(H_0\), \(H_1\) hold, then \({\widehat{\lambda }}<+\infty \).
Proof
Hypotheses \(H_1(i),(ii),(iv)\) imply that we can find \(\lambda _0>0\) such that
Let \(\lambda >\lambda _0\) and suppose that \(\lambda \in {\mathcal {L}}\). Then we can find \(u_\lambda \in S_\lambda \subseteq \mathrm{int \,}C_+\) (see Proposition 6). Let \(\Omega _0\subseteq \Omega \) be an open subset with \(C^2\)-boundary \(\partial \Omega _0\) and such that \({\overline{\Omega }}_0 \subseteq \Omega \). We define
For \(\varepsilon >0\), let \(m_0^\varepsilon =m_0+\varepsilon \). Also, let \(\rho = \max \{ \Vert u_\lambda \Vert _\infty ,m_0^\varepsilon \}\) and take \({\widehat{\xi }}_\rho >0\) as postulated by hypothesis \(H_1(iv)\). We have
For \(\varepsilon >0\) small, we have
So, from (50) and Proposition 2.3 of [13] (see also Proposition A4 of [19]), we obtain
a contradiction. Therefore \({\widehat{\lambda }}\le \lambda _0 < +\infty \). \(\square \)
If \(\lambda \in (0,{\widehat{\lambda }})\), then we have multiplicity of positive solutions.
Proposition 11
If hypotheses \(H_0\), \(H_1\) hold and \(\lambda \in (0,{\widehat{\lambda }})\), then problem (\(P_\lambda \)) has at least two positive solutions \(u_0, {\widehat{u}} \in \mathrm{int \,}C_+\).
Proof
Let \(\beta \in (\lambda ,{\widehat{\lambda }})\) and \(\sigma \in (0,\lambda )\) small such that \(\Vert {\overline{u}}_\sigma \Vert _\infty \le \delta \) (see Proposition 5). From the previous results, we know that for \(u_\beta \in S_\beta \subseteq \mathrm{int \,}C_+\), we can find \(u_0 \in W_0^{1,p(z)}(\Omega )\) such that
As in the proof of Proposition 7, truncating the reaction at \(\{{\overline{u}}_\sigma (z),u_\beta (z)\}\) (see (42)) and introducing the corresponding \(C^1\)-energy functional \({\widehat{\psi }}_\lambda (\cdot )\), via the direct method of the Calculus of Variations, we produce \(u_0\) a global minimizer of \({\widehat{\psi }}_\lambda (\cdot )\).
Also, we introduce the following Carathéodory function
We set \({\widehat{E}}_\lambda (z,x)=\int _0^x {\widehat{e}}_\lambda (z,s)ds\) and introduce the \(C^1\)-functional \({\widehat{\varphi }}_\lambda :W_0^{1,p(z)}(\Omega )\rightarrow {\mathbb {R}}\) defined by
From (42) and (52), we see that
Recall that \(u_0 \in \mathrm{int \,}C_+\) is a global minimizer of \({\widehat{\psi }}_\lambda (\cdot )\). Then from (51) it follows that
Using (52) we can easily check that
Then (54) and (52) imply that we may assume that \(K_{{\widehat{\varphi }}_\lambda }\) is finite or otherwise we already have an infinity of positive smooth solutions of (\(P_\lambda \)) and so we are done. So, we have that \(K_{{\widehat{\varphi }}_\lambda }\) is finite and this fact together with (53) and Theorem 5.7.6, p. 449, of Papageorgiou–Rădulescu–Repovš [18] imply that we can find \(\rho \in (0,1)\) small such that
On account of hypothesis \(H_1(ii)\), we see that if \(u \in \mathrm{int \,}C_+\) then
Moreover, using (52) and arguing as in the proof of Proposition 7 (see the “Claim”), we show that
Then (55), (56) and (57) permit the use of the mountain pass theorem. We can find \({\widehat{u}}\in W_0^{1,p(z)}(\Omega )\) such that
So, \({\widehat{u}}\ne u_0\), \({\widehat{u}}\ne 0\) and \({\widehat{u}}\in \mathrm{int \,}C_+\) is the second positive solution of problem (\(P_\lambda \)) with \(\lambda \in (0,{\widehat{\lambda }})\). \(\square \)
Finally we check the admissibility of the critical parameter \({\widehat{\lambda }}>0\).
Proposition 12
If hypotheses \(H_0\), \(H_1\) hold, then \({\widehat{\lambda }} \in {\mathcal {L}}\).
Proof
Let \(\{ \lambda _n \}_{n \in {\mathbb {N}}}\subseteq {\mathcal {L}}\) be such that \(\lambda _n \uparrow {\widehat{\lambda }}\). We can find \(u_n \in S_{\lambda _n}\subseteq \mathrm{int \,}C_+\) which are minimizers of \({\widehat{\psi }}_{\lambda _n}(\cdot )\) (truncation at \({\overline{u}}_\sigma \) for \(\sigma \in (0,\lambda _n)\) small and at \(u_\beta \in S_{\beta }\subseteq \mathrm{int \,}C_+\) with \(\beta \in (\lambda _n,{\widehat{\lambda }} )\)) and so
Also, we have \({\widehat{\varphi }}_{\lambda _n}^\prime (u_n)=0\) in \(W^{-1,p^\prime (z)}(\Omega )\) for all \(n \in {\mathbb {N}}\). Then as in the proof of Proposition 7 (see the “Claim”), we obtain
We have
Also we have \({\overline{u}}_\sigma \le u_n\) for all \(n \in {\mathbb {N}}\) and so \({\overline{u}}_\sigma \le u_*\) which means that \(u_*\in S_{{\widehat{\lambda }}}\subseteq \mathrm{int \,}C_+\), hence \({\widehat{\lambda }}\in {\mathcal {L}}\). \(\square \)
We have proved that
We can state the following global existence and multiplicity theorem of problem (\(P_\lambda \)) (bifurcation-type theorem).
Theorem 1
If hypotheses \(H_0\), \(H_1\) hold, then there exists \({\widehat{\lambda }}>0\) such that:
-
(a)
for all \(\lambda \in (0,{\widehat{\lambda }})\) problem (\(P_\lambda \)) has at least two positive solutions \(u_0,{\widehat{u}} \in \mathrm{int \,}C_+\);
-
(b)
for \(\lambda ={\widehat{\lambda }}\) problem (\(P_\lambda \)) has at least one positive solution \(u_*\in \mathrm{int \,}C_+\);
-
(c)
for all \(\lambda >{\widehat{\lambda }}\) problem (\(P_\lambda \)) has no positive solution.
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The authors wish to thank the two knowledgeable referees for their corrections and remarks. The second author is supported by the research fund of University of Palermo: “FFR 2023 Calogero Vetro”.
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Papageorgiou, N.S., Vetro, C. & Vetro, F. Singular Anisotropic Problems with Competition Phenomena. J Geom Anal 33, 173 (2023). https://doi.org/10.1007/s12220-023-01227-8
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DOI: https://doi.org/10.1007/s12220-023-01227-8
Keywords
- Singular and concave–convex nonlinearities
- Anisotropic regularity
- Hardy’s inequality
- Global existence and multiplicity of solutions
- Strong comparison
- Truncations