On neumann problems for semilinear elliptic equations with critical nonlinearity:existance and symmetry of multi—peaked solutions1

Abstract

In this paper we construct multi—peaked solutions of a semi—linear elliptic Neumann problem with homogeneous and critical nonlinearity. The multi—peakedness of our solutions is forced by symmetries of the domain. In fact we obain our solutions as local minimizers ofan energy functional in a certain fixed—point space determined by the given symmetries. We are also able to give asymptotic properties on the shape of the solutions as well as on the energy. One particular aspect of our investigations is to determine the exact symmetry of the solutions, i.e.,the isotropy subgroup of our solutions. We find, for instance, all exceptional subgroups of O(3) and the hypercube group in O(n) as isotropy subgroups.