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 $s_g$ 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.