Effect of symmetry to the structure of positive solutions in nonlinear elliptic problems, II

Abstract

We consider the problem; Deltau + hu + f(u) = 0 in Omega (R) u = 0 on partial derivative Omega (R) u > 0 in Omega (R) where Q(R) equivalent to \x is an element of 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. Let G be an infinite closed subgroup of O(N). We investigate how the symmetry subgroup G affects the structure of positive solutions. Considering a natural G group action on a sphere SN-1 we give a partial order on the space of G-orbits {xG \ x is an element of SN-1}. In a previous paper. we studied the effect of symmetry on the structure of positive solutions when the number of elements of xG is finite for some x is an element of SN-1. In this paper, we study the effect when re is an infinite set for any x is an element of SN-1. In fact, in view of the partial order, a critically (locally minimal) orbital set will be defined. Then. it is shown that. when R --> proportional to a critical orbital set produces a solution of our problem whose energy goes to proportional to and is concentrated around the scaled critical orbital set. (C) 2001 Academic Press.