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Towers of Bubbles for Yamabe-Type Equations and for the Brézis–Nirenberg Problem in Dimensions \(n \ge 7\)

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Abstract

Let (M, g) be a closed locally conformally flat Riemannian manifold of dimension \(n \ge 7\) and of positive Yamabe type. If \(h \in C^1(M)\) and \(\xi _0\) is a non-degenerate critical point of the mass function we prove the existence, for any \( k \ge 1\) of a positive blowing-up solution \(u_\varepsilon \) of

$$\begin{aligned} \triangle _g u_\varepsilon +\big ( c_n S_g +\varepsilon h\big ) u_\varepsilon = u_\varepsilon ^{2^*-1} \end{aligned}$$

that blows up, as \(\varepsilon \rightarrow 0\), like the superposition of k positive bubbles concentrating at different speeds at \(\xi _0\). The method of proof combines a finite-dimensional reduction with the sharp pointwise analysis of solutions of a linear problem. As another application of this method of proof we construct sign-changing blowing-up solutions \(u_\varepsilon \) for the Brézis–Nirenberg problem

$$\begin{aligned} \triangle _{\xi } u_\varepsilon - \varepsilon u_\varepsilon = |u_\varepsilon |^{\frac{4}{n-2}} u_\varepsilon \text { in } \Omega , \quad u_\varepsilon = 0 \text { on } \partial \varOmega \end{aligned}$$

on a smooth bounded open set \(\varOmega \subset {\mathbb {R}}^n\), \(n \ge 7\), that look like the superposition of k positive bubbles of alternating sign as \(\varepsilon \rightarrow 0\).

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Acknowledgements

The author warmly thanks A. Pistoia for many fruitful and motivating discussions during the preparation of this work, and in particular for pointing out the possible application to the Brézis–Nirenberg problem, which led to Theorem 1.2. The author was supported by a FNRS CdR Grant J.0135.19 and by the Fonds Thélam.

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  1. Université Libre de Bruxelles, Service d’Analyse CP 214, Boulevard du Triomphe, 1050, Brussels, Belgium

    Bruno Premoselli

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Appendix A: Technical Results

Appendix A: Technical Results

We keep the notations of Sect. 2. The integer \(k\ge 1\) refers to the number of bubbles.

1.1 A.1. Proof of (2.21) and (2.22)

Let \(1 \le i < j \le k\) and \(0 \le p,q \le n\). We first prove (2.21). We write for this, using (2.14), (2.19) and (2.20) and since \(|Z_{i,p}| \lesssim W_i\), that

$$\begin{aligned} \begin{aligned} \langle Z_{i,p}, Z_{i,q}\rangle&= \int _M \big ( \triangle _g + c_n S_g + \varepsilon h) Z_{i,p} Z_{i,q}\mathrm{d}v_g \\&= (2^*-1) \int _M W_i^{2^*-2} Z_{i,p} Z_{i,q} \mathrm{d }v_g + O(\int _M W_i^2 \mathrm{d}v_g) \\&= (2^*-1) \int _M W_i^{2^*-2} Z_{i,p} Z_{i,q} \mathrm{d}v_g + O(\mu _i^2). \end{aligned} \end{aligned}$$

Letting \(y = \exp _{\xi _i}^{g_{\xi _i}}(\mu _i x)\) and using the conformal invariance of the conformal laplacian and (2.4) we get:

$$\begin{aligned}\begin{aligned} \int _M W_i^{2^*-2} Z_{i,p} Z_{i,q} \mathrm{d}v_g&= \int _{B_{g_{\xi _i}}(\xi _i, r_0)} W_i^{2^*-2} Z_{i,p} Z_{i,q} \mathrm{d}v_g + O(\mu _i^n) \\&= \int _{B\left( 0, \frac{r_0}{\mu _i}\right) } U_0^{2^*-2} V_p V_q \mathrm{d}x + O(\mu _i^4) \\&= \int _{{\mathbb {R}}^n} U_0^{2^*-2} V_p V_q \mathrm{d}x + O(\mu _i^4) \\&= \Vert \nabla V_p \Vert _{L^2({\mathbb {R}}^n)}^2 \delta _{pq}+ O(\mu _i^4), \end{aligned} \end{aligned}$$

where \(r_0\) is as in (2.3), which proves (2.21).

We now prove (2.22). With (2.19) and (2.20) we have:

$$\begin{aligned} \begin{aligned}&\langle Z_{i,p}, Z_{j,q} \rangle \\&\quad = \int _M \Big [ \triangle _gZ_{j,q} + (c_n S_g + \varepsilon h) Z_{j,q} \Big ] Z_{i,p} \mathrm{d}v_g \\&\quad = (2^*-1) \int _M W_j^{2^*-2} Z_{j,q}Z_{i,p} \mathrm{d}v_g+\int _M \Big ( \varepsilon Z_{j,q}Z_{i,p} + O \big (\mu _j W_j |Z_{i,q}|\big ) \Big ) \mathrm{d}v_g \\&\quad = O \left( \left( \frac{\mu _j}{\mu _i} \right) ^{\frac{n-2}{2}} \right) , \end{aligned} \end{aligned}$$

where we just used the rough estimate \(|Z_{i,p}| \lesssim \mu _i^{1 - \frac{n}{2}}\).

1.2 A.2. Additional Material

We prove two inequalities that were used several times throughout the proof.

Lemma A.1

Let \(i \in \{1, \dots , k\},\) and \(x \in M\). We have:

$$\begin{aligned} \begin{aligned}&\text { If } 0 \le p < n-4: \\&\int _{B_i\backslash B_{i+1}} d_g(x, \cdot )^{2-n} W_i^{2^*-2} \left( \frac{\mu _i}{\theta _i} \right) ^p \mathrm{d}v_g \\&\quad \lesssim \left\{ \begin{aligned}&\left( \frac{\mu _i}{\theta _i(x)} \right) ^{p+2}&\text { if } x \in B_i, \\&\mu _i^{\frac{p+2}{2}} \mu _{i-1}^{\frac{n-4-p}{2}} W_i (x)&\text { if } x \in M \backslash B_i, \end{aligned} \right. \\&\text { If } p > n-4: \\&\int _{B_i\backslash B_{i+1}} d_g(x, \cdot )^{2-n} W_i^{2^*-2} \left( \frac{\mu _i}{\theta _i} \right) ^p \mathrm{d}v_g \lesssim \left( \frac{\mu _i}{\theta _i(x)} \right) ^{n-2} = \mu _i^\frac{n-2}{2}W_i(x) , \\ \end{aligned} \end{aligned}$$
(A.1)

where \(\theta _i \) is as in (2.18).

Proof

Assume first that \(d_{g_{\xi _i}}(\xi _i, x) \le 2 \sqrt{\mu _i \mu _{i-1}}\). Then \(W_i^{2^*-2} \lesssim \mu _i^2 \theta _i(x)^{-4}\), where \(\theta _i\) is as in (2.18). If \(0 \le p < n-4\), the result follows from Giraud’s lemma. While if \(p > n-4\) letting \(y = \exp _{\xi _i}^{g_{\xi _i}}(\mu _i z)\) in the integral and \({\check{x}} = \frac{1}{\mu _i} {\exp _{\xi _i}^{g_{\xi _i}}}^{-1}(x)\) yields:

$$\begin{aligned}\begin{aligned} \int _{B_i\backslash B_{i+1}} d_g(x, \cdot )^{2-n} W_i^{2^*-2} \left( \frac{\mu _i}{\theta _i} \right) ^p\mathrm{d}v_g&\lesssim \int _{{\mathbb {R}}^n} |{\check{x}}-z|^{2-n} (1+|z|)^{-n-2}\mathrm{d}z \\&\lesssim \frac{1}{(1+|{\check{x}}|)^{n-2}} \lesssim \left( \frac{\mu _i}{\theta _i(x)} \right) ^{n-2}. \end{aligned} \end{aligned}$$

Assume then that \(d_{g_{\xi _i}}(\xi _i, x) \ge 2 \sqrt{\mu _{i-1}\mu _i}\) (note that this only makes sense for \( i\ge 2\)). Then for any \(y \in B_i\) one has \(d_g(x,y) > rsim d_g(x, \xi _i) > rsim \theta _i(x)\). If \(p < n-4\) we then have

$$\begin{aligned} \begin{aligned} \int _{B_i\backslash B_{i+1}} d_g(x, \cdot )^{2-n} W_i^{2^*-2} \left( \frac{\mu _i}{\theta _i} \right) ^p\mathrm{d}v_g&\lesssim \theta _i(x)^{2-n} \int _{B_i} W_i^{2^*-2} \left( \frac{\mu _i}{\theta _i} \right) ^p\\&= \theta _i(x)^{2-n} \mu _i^{p+2} (\mu _i \mu _{i-1})^{\frac{n-4-p}{2}} \\&= \mu _i^{\frac{p+2}{2}} \mu _{i-1}^{\frac{n-4-p}{2}} W_i (x), \end{aligned} \end{aligned}$$

while if \(p > n-4\) we have

$$\begin{aligned} \begin{aligned} \int _{B_i\backslash B_{i+1}} d_g(x, \cdot )^{2-n} W_i^{2^*-2} \left( \frac{\mu _i}{\theta _i} \right) ^p\mathrm{d}v_g&\lesssim \theta _i(x)^{2-n} \int _{B_i} W_i^{2^*-2} \left( \frac{\mu _i}{\theta _i} \right) ^p\\&= \theta _i(x)^{2-n} \mu _i^{n-2} \int _{{\mathbb {R}}^n} (1+|y|)^{-p-4} \mathrm{d}y \\&\lesssim \left( \frac{\mu _i}{\theta _i(x)}\right) ^{n-2}. \end{aligned} \end{aligned}$$

Note finally that, in the intermediate case \(\sqrt{\mu _{i-1}\mu _i} \le d_{g_{\xi _i}}(\xi _i, x) \le 2 \sqrt{\mu _{i-1}\mu _i}\) we have

$$\begin{aligned} \left( \frac{\mu _i}{\theta _i(x)} \right) ^{p+2} \sim \left( \frac{\mu _i}{\mu _{i-1}}\right) ^{\frac{p+2}{2}} \sim \mu _i^{\frac{p+2}{2}} \mu _{i-1}^{\frac{n-4-p}{2}} W_i (x) \end{aligned}$$

and this holds true for \(0 \le p \le n-4\), so that the result follows. \(\square \)

Lemma A.2

For any \(i, j \in \{1, \dots , k\}\), \(i < j,\) and any \(x \in M,\) we have

$$\begin{aligned} \begin{aligned}&\int _{B_{i} \backslash B_{i+1}} W_i^{2^*-2} W_j d_g(x,\cdot )^{2-n}\mathrm{d}v_g \\&\quad \lesssim \left\{ \begin{array}{ll} \left( \frac{\mu _j}{\mu _{i+1}} \right) ^{\frac{n-2}{2}} \frac{\mu _{i+1}}{\mu _i} \mu _i^{1 - \frac{n}{2}} &{} \text {if } x \in B_{i+1}, \\ \left( \frac{\mu _j}{\mu _i} \right) ^{\frac{n-2}{2}} W_i(x) &{} \text {if } x \in B_i \backslash B_{i+1} \text { and } \theta _j(x) \ge \mu _i , \\ \frac{\theta _j(x)^2}{\mu _{i}^2} W_j(x) &{} \text {if } x \in B_i \backslash B_{i+1} \text { and } \theta _j(x) \le \mu _i ,\\ \left( \frac{\mu _j}{\mu _i} \right) ^{\frac{n-2}{2}} W_i(x) &{} \text {if } x \in M \backslash B_i . \end{array} \right. \\ \end{aligned} \end{aligned}$$
(A.2)

Proof

Assume first that \(d_{g_{\xi _i}}(\xi _i, x) \ge \sqrt{\mu _i \mu _{i-1}}\), that is \(x \in M \backslash B_i\) (this case is empty if \(i=1\)). Changing variables in the integral by \(y = \exp _{\xi _i}^{g_{\xi _i}}(\mu _i x)\) and using Giraud’s lemma yields in this case

$$\begin{aligned} \int _{B_{i} \backslash B_{i+1}} W_i^{2^*-2} W_j d_g(x,\cdot )^{2-n} \mathrm{d}v_g \lesssim \theta _i(x)^{2-n} \mu _j^{\frac{n-2}{2}} = \left( \frac{\mu _j}{\mu _i} \right) ^{\frac{n-2}{2}} W_i(x) . \end{aligned}$$

Assume now that \(x \in B_{i+1}\), so in particular \(\theta _j(x) \lesssim \sqrt{\mu _{i+1} \mu _i}\) by (2.18). We let the change of variables

$$\begin{aligned} y = \exp _{\xi _j}^{g_{\xi _j}}(\theta _j(x) z) \end{aligned}$$

in the integral. By (2.10) one has

$$\begin{aligned} \begin{aligned}&\int _{B_{i} \backslash B_{i+1}} W_i^{2^*-2} W_j d_g(x,\cdot )^{2-n}\mathrm{d}v_g \\&\quad \lesssim \mu _i^{-2} \mu _j^{\frac{n-2}{2}} \theta _j(x)^{4-n} \\&\qquad \times \int _{B_0 \left( 2\frac{\sqrt{\mu _{i-1}\mu _i}}{\theta _j(x)} \right) \backslash B_0 \left( \frac{1}{2} \frac{\sqrt{\mu _{i+1}\mu _i}}{\theta _j(x)} \right) } |{\check{x}}_i - z|^{2-n} \Big ( \frac{\mu _j}{\theta _j(x)} + |z|\Big ) ^{2-n} \mathrm{d}z, \end{aligned} \end{aligned}$$

where we have let \({\check{x}}_i= \frac{1}{\theta _j(x)} (\exp _{\xi _j}^{g_{\xi _j}})^{-1}(x)\). As is easily checked, the previous integral is always uniformly integrable whatever the value of \(\theta _j(x)\) is, and there holds \(|{\check{x}}_i| \le 1\) by definition of \(\theta _j\). So that in the end

$$\begin{aligned} \begin{aligned} \int _{B_{i} \backslash B_{i+1}} W_i^{2^*-2} W_j d_g(x,\cdot )^{2-n} \mathrm{d}v_g&\lesssim \mu _i^{-2} \mu _j^{\frac{n-2}{2}} \theta _j(x)^{4-n} \int _{{\mathbb {R}}^n \backslash B_0 \left( \frac{\sqrt{\mu _{i+1}\mu _i}}{\theta _j(x)} \right) } |z|^{4-2n} \mathrm{d}z \\&\lesssim \mu _i^{-2} \mu _j^{\frac{n-2}{2}} \theta _j(x)^{4-n}\left( \frac{\theta _j(x)}{\sqrt{\mu _{i+1}\mu _i}}\right) ^{n-4} \\&\lesssim \left( \frac{\mu _j}{\mu _{i+1}} \right) ^{\frac{n-2}{2}} \frac{\mu _{i+1}}{\mu _i} \mu _i^{1 - \frac{n}{2}}. \end{aligned} \end{aligned}$$

Assume now that \(x \in B_i \backslash B_{i+1}\). The change of variables

$$\begin{aligned} y = \exp _{\xi _j}^{g_{\xi _j}}(\mu _i z) \end{aligned}$$

gives

$$\begin{aligned} \begin{aligned}&\int _{B_{i} \backslash B_{i+1}} W_i^{2^*-2} W_j d_g(x,\cdot )^{2-n} \mathrm{d}v_g \\&\quad \lesssim \mu _j^{\frac{n-2}{2}} \mu _i^{2-n} \int _{{\mathbb {R}}^n \backslash B_0 \big (\frac{1}{2} \sqrt{\frac{\mu _{i+1}}{\mu _i}}\big )} (1 + |{\check{\xi }}-y|)^{-4} |y|^{2-n} |{\check{x}}-y|^{2-n} \mathrm{d}y, \end{aligned} \end{aligned}$$
(A.3)

where we have let \({\check{\xi }} = \frac{1}{\mu _i} (\exp _{\xi _j}^{g_{\xi _j}})^{-1}(\xi _i)\) and \({\check{x}} = \frac{1}{\mu _i} (\exp _{\xi _j}^{g_{\xi _j}})^{-1}(x)\). Since \(x \not \in B_{i+1}\) we have \(\theta _j(x) > rsim \sqrt{\mu _{i+1} \mu _i}\) and thus \(|{\check{x}}| > rsim \frac{\theta _j(x)}{\mu _i}\). Assume first that \(|{\check{x}}| > rsim 1\). Then we also have \(\theta _j(x) > rsim \theta _i(x)\), so that classical integral comparison results yield

$$\begin{aligned} \int _{B_{i} \backslash B_{i+1}} W_i^{2^*-2} W_j d_g(x,\cdot )^{2-n} \mathrm{d}v_g \lesssim \mu _j^{\frac{n-2}{2}} \mu _i^{2-n} (1 + |{\check{x}}|)^{2-n} \lesssim \left( \frac{\mu _j}{\mu _i} \right) ^{\frac{n-2}{2}} W_i(x) . \end{aligned}$$

Assume finally that \(|{\check{x}}| \lesssim 1\). A standard Giraud-type argument (see e.g. Lemma 7.5 in Hebey [14]) yields with (A.3)

$$\begin{aligned} \begin{aligned} \int _{B_{i} \backslash B_{i+1}} W_i^{2^*-2} W_j d_g(x,\cdot )^{2-n} \mathrm{d}v_g&\lesssim \mu _j^{\frac{n-2}{2}} \mu _i^{2-n} |{\check{x}}|^{4-n} \\&\lesssim \frac{\theta _j(x)^2}{\mu _{i}^2} W_j(x), \end{aligned} \end{aligned}$$

which concludes the proof of (A.2). \(\square \)

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Premoselli, B. Towers of Bubbles for Yamabe-Type Equations and for the Brézis–Nirenberg Problem in Dimensions \(n \ge 7\). J Geom Anal 32, 73 (2022). https://doi.org/10.1007/s12220-021-00836-5

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