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.


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


Zbl 0952.35054
Full Text: DOI


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