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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

Citations:

Zbl 0952.35054
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References:

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