Multipeak Solutions for the Yamabe Equation

Abstract

Let (M, g) 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 (X, h) is a Riemannian manifold with constant positive scalar curvature. We also prove that solutions with small energy only have one local maximum.

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 $m+n\le 9$.