Effect of Symmetry to the Structure of Positive Solutions in Nonlinear Elliptic Problems, II

Abstract

We consider the problem:Δu+hu+f(u)=0inΩRu=0on∂ΩRu>0inΩR, where ΩR≡{x∈RN∣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∈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∈SN−1. In this paper, we study the effect when xG is an infinite set for any x∈SN−1. In fact, in view of the partial order, a critical (locally minimal) orbital set will be defined. Then, it is shown that, when R→∞ a critical orbital set produces a solution of our problem whose energy goes to ∞ and is concentrated around the scaled critical orbital set.