Multipeak solutions for the Yamabe equation

Carolina A. Rey, Juan Miguel Ruiz

Published 2018-07-22Version 1

Let $(M,g)$ be a closed Riemannian manifold of dimension $n\geq 3$ and $x_0 \in M$ be an isolated local maximum 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 $-\epsilon^2 \Delta u +(1+ \epsilon^2 \lambda s_g) u = u^q$ has a positive $k$-peaks solution which concentrate around $x_0$, assuming that a constant $\beta$ is non-zero. In the equation $\lambda \in {\mathbb R}$ is any constant and we will assume that $q<\frac{n+2}{n-2}$. The constant $\beta$ depends on the dimension and $q$, and can be easily computed numerically, being positive in all cases considered. This provides solutions to the Yamabe equation on Riemannian products $(M\times N, g+ \epsilon h)$, where $(N,h)$ is a Riemannian manifold with constant positive scalar curvature. We also prove that solutions with small energy only have one local maximum.