Multipeak solutions for the Yamabe equation

TL;DR

This paper constructs multiple-peaks solutions to a perturbed Yamabe equation on Riemannian manifolds, showing concentration around minima of scalar curvature and analyzing the solutions' maximum points.

Contribution

It introduces a method to find multi-peak solutions concentrating near scalar curvature minima for small perturbations of the Yamabe equation.

Findings

Existence of k-peak solutions concentrating near scalar curvature minima.

Solutions with small energy have only one local maximum.

Application to Riemannian products with constant positive scalar curvature.

Abstract

Let $(M,g)$ be a closed Riemannian manifold of dimension $n\geq 3$ and $x_0 \in M$ be an isolated local minimum of the scalar curvature $s_g$ of $g$. For any positive integer $k$ we prove that for $\epsilon >0$ small enough the subcritical Yamabe equation $-\epsilon^2 \Delta u +(1+ c_{N} \ \epsilon^2 s_g ) u = u^q$ 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.