Multipeak Solutions for the Yamabe Equation

Abstract

Let (Mg) be a closed Riemannian manifold of dimension \(n\ge 3\) and \(x_0 \in M\) be an isolated local minimum of the scalar curvature \(s_\mathrm{{g}}\) of g. For any positive integer k we prove that for \(\epsilon >0\) small enough the subcritical Yamabe equation \(-\epsilon ^2 \Delta u +(1+ c_{N} \ \epsilon ^2 s_\mathrm{{g}}) u = u^\mathrm{{q}}\) has a positive k-peaks solution which concentrate around \(x_0\), assuming that a constant \(\beta \) is non-zero. In the equation \(c_N = \frac{N-2}{4(N-1)}\) for an integer \(N>n\) and \(q= \frac{N+2}{N-2}\). The constant \(\beta \) depends on n and N, and can be easily computed numerically, being negative in all cases considered. This provides solutions to the Yamabe equation on Riemannian products \((M\times X , g+ \epsilon ^2 h )\), where (Xh) is a Riemannian manifold with constant positive scalar curvature. We also prove that solutions with small energy only have one local maximum.

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Acknowledgements

The authors would like to thank Prof. Jimmy Petean for many helpful discussions on the subject. The second author was supported by program UNAM-DGAPA-PAPIIT IA106918.

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Correspondence to Juan Miguel Ruiz.

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Appendix

Appendix

In this section we compute numerically \(\beta \) of Sect. 3, for low dimensions. Namely

$$\begin{aligned} \beta := \mathbf{c} \displaystyle \int _{\mathbb {R}^n} U^2(z) \ \mathrm{{d}}z - \frac{1}{n(n+2)}\displaystyle \int _{\mathbb {R}^n} |\nabla U(z)|^2 |z|^2 \ \mathrm{{d}}z, \end{aligned}$$
(108)

which plays an important role in the asymptotic expansion of the energy \(\overline{J_{\epsilon }}\).

Table 1 Numerical values for \(\beta \), for \(n+m \le 9\)

\(\beta \) is a dimensional constant that requires knowledge of the unique (up to translations) positive solution \(U \in H^1(\mathbb {R}^n)\) that vanishes at infinity of

$$\begin{aligned} -\Delta U + U=U^{p-1} \hbox { in } \mathbb {R}^n, \end{aligned}$$
(109)

with \(p=\frac{2(m+n)}{m+n-2}\). The solution is known to exist, and to be unique and radial, see [14] for details.

Hence, we consider the solution \(h=h_{\alpha }\) of

$$\begin{aligned} h''(t)+\frac{n-1}{t}h'(t)-h(t)+h(t)^{\frac{m+n+2}{m+n-2}}=0 \end{aligned}$$
(110)

with \(h(0)=\alpha >0\), \(h'(0)=0\). By the aforementioned existence and uniqueness results, there exists only one value \(\alpha =\alpha _0=\alpha _0(m,n)\) that gives a positive solution \(h_{\alpha _0}\) that vanishes at infinity. Our approach is to find \(h_{\alpha _0}\) numerically as the solution of (110) that vanishes at infinity, and then to integrate it numerically to find \(V_{n-1} \ \mathbf{c} \int _{0}^{\infty } u^2 r^{n-1}dr\) and \(\frac{V_{n-1}}{n(n+2)} \int _{0}^{\infty } u'^2 r^{n+1} dr \), the two terms involved in (108). Of course \(u(r)=h_{\alpha _0}(r)\).

In Table 1 we show the numerical results, where \(\beta \) is negative for \(m+n\le 9\).

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Rey, C.A., Ruiz, J.M. Multipeak Solutions for the Yamabe Equation. J Geom Anal 31, 1180–1222 (2021). https://doi.org/10.1007/s12220-019-00258-4

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Keywords

  • Yamabe problem
  • Elliptic PDE on manifolds
  • Scalar curvature
  • Finite dimensional reduction

Mathematics Subject Classification

  • 35J60
  • 58J05
  • 35B33
  • 35B09