## Advances in Differential Equations

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

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

#### 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.