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Effect of symmetry to the structure of positive solutions in nonlinear elliptic problems. III. (English) Zbl 1221.35142

Summary: We consider the problem: \[ \begin{cases} \Delta u+ u^p=0 \quad &\text{in } \;\Omega, \\ u=0\quad &\text{on} \partial \;\Omega, \\ u>0\quad &\text{in} \;\Omega, \end{cases} \] , where \(\Omega=\{x\in {\mathbb R}^N \colon |R-1|<|x|<R+1 \}\) 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 of \(O(N)\). In the first part [J. Byeon, J. Differ. Equations 163, No. 2, 429–474 (2000; Zbl 0952.35054)], an existence of locally minimal energy solutions due to a structural property of the orbits space was shown. In this paper, it will be showed that more various types of solutions than those obtained in [loc. cit.], which are close to a finite sum of locally minimal energy solutions. Furthermore, we discuss possible types of solutions and show that any solution with exactly two local maximum points should symmetric.
For Part II, cf. [J. Differ. Equations 173, No. 2, 321–355 (2001; Zbl 0989.35053)].

MSC:

35J60 Nonlinear elliptic equations
35A30 Geometric theory, characteristics, transformations in context of PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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