## Advances in Differential Equations

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

Jun Hirata

#### 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

#### 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