Nonradial Clustered Spike Solutions for Semilinear Elliptic Problems on S

We consider the following superlinear elliptic equation on S ε∆Snu− u + u = 0 in S, u > 0 in S where ∆Sn is the Laplace-Beltrami operator on S. We prove that for any k = 1, ..., n−1, there exists pk > 1 such that for 1 < p < pk and ε sufficiently small, there exist at least n − k positive solutions concentrating on k−dimensional subset of the equator. We also discuss the problem on geodesic balls of S and establish the existence of positive nonradial solutions. The method extends to Dirichlet problems with more general nonlinearities. The proofs are based on the finite-dimensional reduction procedure which was successfully used by the second author in singular perturbation problems.