Multipeak Solutions for the Yamabe Equation

Abstract

Let (M, g) be a closed Riemannian manifold of dimension n≥ 3 and x∈ M be an isolated local minimum of the scalar curvature sg of g. For any positive integer k we prove that for ϵ> 0 small enough the subcritical Yamabe equation -ϵ2Δu+(1+cNϵ2sg)u=uq has a positive k-peaks solution which concentrate around x, assuming that a constant β is non-zero. In the equation cN=N-24(N-1) for an integer N> n and q=N+2N-2. The constant β 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× X, g+ ϵ2h) , 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.