## Advances in Differential Equations

- Adv. Differential Equations
- Volume 16, Number 3/4 (2011), 289-324.

### $G$-invariant positive solutions for a quasilinear Schrödinger equation

Shinji Adachi and Tatsuya Watanabe

#### Abstract

We are concerned with a quasilinear elliptic equation of the form \begin{equation*} -\Delta u+a(x) u-\Delta(|u|^\alpha)|u|^{\alpha-2}u=h(u)\quad \hbox{in }\mathbb R^N, \end{equation*} where $\alpha> 1$ and $N\geq 1$. By using variational approaches, we prove the existence of at least one positive solution of the above equation under suitable conditions on $a(x)$ and $h$. In particular, we are interested in the situation that $a(x)$ is invariant under the finite group action $G$.

#### Article information

**Source**

Adv. Differential Equations Volume 16, Number 3/4 (2011), 289-324.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355854310

**Mathematical Reviews number (MathSciNet)**

MR2767080

**Subjects**

Primary: 35J60: Nonlinear elliptic equations 58E4

#### Citation

Adachi, Shinji; Watanabe, Tatsuya. $G$-invariant positive solutions for a quasilinear Schrödinger equation. Adv. Differential Equations 16 (2011), no. 3/4, 289--324. https://projecteuclid.org/euclid.ade/1355854310.