## Effect of symmetry to the structure of positive solutions in nonlinear elliptic problems. II.(English)Zbl 0989.35053

Summary: The author considers the problem: \begin{aligned} \Delta u+hu+ f(u)=0 & \text{ in }\Omega_R\\ u=0 & \text{ on }\partial \Omega_R\\ u>0 & \text{ in }\Omega_R, \end{aligned} where $$\Omega_R\equiv \{x\in\mathbb{R}^N \mid R-1<|x |< R+1\}$$ 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 $$O(N)$$. He investigates how the symmetry subgroup $$G$$ affects the structure of positive solutions. Considering a natural $$G$$ group action on a sphere $$S^{N-1}$$, we give a partial order on the space of $$G$$-orbits $$\{xG\mid x\in S^{N-1}\}$$. In a previons paper [J. Byeon, J. Differ. Equations 163, No. 2, 429-474 (2000; Zbl 0952.35054)], the author has studied the effect of symmetry on the structure of positive solutions when the number of elements of $$xG$$ is finite for some $$x\in S^{N-1}$$. In this paper, he studies the effect when $$xG$$ is an infinite set for any $$x\in S^{N-1}$$. In fact, in view of the partial order, a critical (locally minimal) orbital set will be defined. Then it is shown that when $$R\to\infty$$, a critical orbital set produces a solution of our problem whose energy goes to $$\infty$$ and is concentrated around the scaled critical orbital set.

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35B40 Asymptotic behavior of solutions to PDEs

### Keywords:

locally minimal; energy solution; critical orbital set

Zbl 0952.35054
Full Text:

### References:

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