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
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
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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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
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:
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:
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:
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:
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
while if \(p > n-4\) we have
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
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
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
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
in the integral. By (2.10) one has
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
Assume now that \(x \in B_i \backslash B_{i+1}\). The change of variables
gives
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
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)
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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DOI: https://doi.org/10.1007/s12220-021-00836-5