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2007 A positive solution of a nonlinear elliptic equation in $\Bbb R^N$ with $G$-symmetry
Jun Hirata
Adv. Differential Equations 12(2): 173-199 (2007).

Abstract

In this paper we consider the following elliptic equation: \begin{equation*} \begin{cases} \; -\Delta u + u = f(x,u) \qquad \text{in}\; \mathbf{R}^N, \\ \quad u \in H^1(\mathbf{R}^N). \end{cases} \end{equation*} Here $f(x,u)$ is invariant under a finite group action $G \subset O(N)$ which acts on $S^{N-1}$ effectively. When $N \geq 3$, we show the existence of a positive solution without global assumptions like: $ u \mapsto \frac{f(x,u)}{u} $ is increasing in $ u > 0$, $f(x,u) \geq f^{\infty}(u)$. We can deal with asymptotically linear equations as well as superlinear equations. Interaction estimates between solution at infinity play important roles in our argument.

Citation

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Jun Hirata. "A positive solution of a nonlinear elliptic equation in $\Bbb R^N$ with $G$-symmetry." Adv. Differential Equations 12 (2) 173 - 199, 2007.

Information

Published: 2007
First available in Project Euclid: 18 December 2012

zbMATH: 1166.35015
MathSciNet: MR2294502

Subjects:
Primary: 35J60
Secondary: 35B05, 47J30, 58E05

Rights: Copyright © 2007 Khayyam Publishing, Inc.

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Vol.12 • No. 2 • 2007
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