ON NEUMANN PROBLEMS FOR SEMILINEAR ELLIPTIC-EQUATIONS WITH CRITICAL NONLINEARITY - EXISTENCE AND SYMMETRY OF MULTI-PEAKED SOLUTIONS

In this paper we construct multi-peaked solutions of a semilinear elli ptic Neumann problem with homogeneous and critical nonlinearity. The m ulti-peakedness of our solutions is forced by symmetries of the domain . In fact we obtain our solutions as local minimizers of an energy fun ctional in a certain fixed-point space determined by the given symmetr ies. We are also able to give asymptotic properties on the shape of th e solutions as well as on the energy. One particular aspect of our inv estigations is to determine the exact symmetry of the solutions, i.e., the isotropy subgroup of our solutions. We find, for instance, all ex ceptional subgroups of O(3) and the hypercube group in O(n) as isotrop y subgroups.