Effect of symmetry to the structure of positive solutions in nonlinear eliptic problems

Abstract

We consider the problem: Delta u + hu + f(u) = 0 in Omega(R) u = 0 on partial derivative Omega(R) u > 0 in Omega(R), where Omega(R) = { x epsilon R-N \ R-1 < \x\ < R + 1}, and the function f and the constant h satisfy suitable assumptions. This problem is invariant under the orthogonal coordinate transformations, in other words, O(N)-symmetric. We investigate how the symmetry affects to the structure of positive solutions. For a closed subgroup G of O(N), we consider a natural group action G x SN-1 --> SN-1. Then, we give a partial order on the space of G-orbits. Then, with respect to the partial order, a critical (locally minimal) orbital set will be defined. As a main result of this paper, we show that, when R --> infinity, a critical orbital set produces a solution of our problem whose energy is concentrated around a scaled critical orbital set. (C) 2000 Academic Press.