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

We consider the problem: Δu+up=0 in ΩR,u=0 on ∂ΩR,u>0 in ΩR, where ΩR≡{x∈RN|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 HGR≡{u∈H01,2(RN) |u(x)=u(gx), x∈ΩR, g∈G}. In the earlier paper [5], an existence of locally minimal energy solutions in HGR due to a structural property of the orbits space of an action G×SN-1→SN-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 HRG 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. [ABSTRACT FROM AUTHOR]