Non-radial clustered spike solutions for semilinear elliptic problems on sN

Abstract

We consider the superlinear elliptic equation on Sn $$\varepsilon ^2 \Delta _{S^n } u - u + u^p = 0 in S^n , u > 0 in S^n ,$$ where ΔSn is the Laplace-Beltrami operator on S n. We prove that for any k = 1,..., n − 1, there exists p k > 1 such that for 1 < p < p k and e sufficiently small, there exist at least n−k positive solutions concentrating on a k-dimensional subset of the equator. We also discuss the problem on geodesic balls of S n and establish the existence of positive non-radial solutions. The method extends to Dirichlet problems with more general non-linearities. The proofs are based on the finite-dimensional reduction procedure which was successfully used by the second author in singular perturbation problems.