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

Abstract

We consider the problem: Δu+u^p=0 in Ω_R,u=0 on ∂Ω_R,u>0 in Ω_R, where Ω_R≡{x∈R^N|R−1<|x|<R+1}, N≥3, and 1<p<(N+2)/(N−2).

This problem is invariant under the orthogonal coordinate transformations, in other words, O(N)‐symmetric. Let G be a closed subgroup O(N), and H^G_R≡{u∈H₀^1,2(R^N) |u(x)=u(gx), x∈Ω_R, g∈G}. In the earlier paper [5], an existence of locally minimal energy solutions in H^G_R due to a structural property of the orbits space of an action G×S^N−1→S^N−1 was showed for large R. In this paper, it will be showed that more various types of solutions than those obtained in [5], which are close to a finite sum of locally minimal energy solutions in H_R^G for some G⊂O(N), appear as R→∞. Furthermore, we discuss possible types of solutions and show that any solution with exactly two local maximum points should be O(N−1)‐symmetric for large R>0.