Advances in Differential Equations

A positive solution of a nonlinear elliptic equation in $\Bbb R^N$ with $G$-symmetry

Jun Hirata

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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.

Article information

Source
Adv. Differential Equations Volume 12, Number 2 (2007), 173-199.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355867474

Mathematical Reviews number (MathSciNet)
MR2294502

Zentralblatt MATH identifier
1166.35015

Subjects
Primary: 35J60: Nonlinear elliptic equations
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 47J30: Variational methods [See also 58Exx] 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)

Citation

Hirata, Jun. A positive solution of a nonlinear elliptic equation in $\Bbb R^N$ with $G$-symmetry. Adv. Differential Equations 12 (2007), no. 2, 173--199. https://projecteuclid.org/euclid.ade/1355867474.


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