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

Abstract

We consider the problem: Deltau+u(p)=0 in Omega(R), u=0 on partial derivativeOmega(R), u>0 in Omega(R), where Omega(R)equivalent to{xis an element ofR(N)|R-1<\x\<R+1}, Ngreater than or equal to3, 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(R)(G)equivalent to{uis an element ofH(0)(1,2)(R-N) |u(x)=u(gx), xis an element ofOmega(R), gis an element ofG}. In the earlier paper [5], an existence of locally minimal energy solutions in H-R(G) due to a structural property of the orbits space of an action GxS(N-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 H-R(G) for some Gsubset ofO(N), appear as R-->infinity. 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.