## Differential and Integral Equations

- Differential Integral Equations
- Volume 23, Number 1/2 (2010), 113-123.

### Existence of radial solutions for a class of $p(x)$-Laplacian equations with critical growth

#### Abstract

Using variational methods we establish the existence of solutions for the following class of $p(x)$-Laplacian equations: \begin{equation} -div(|\nabla u|^{p(x)-2}\nabla u) + |u|^{p(x)-2}u= \lambda |u|^{q(x)-2}u + |u|^{p^{*}(x)-2}u, \,\,\,\,\, \mathbb{R}^{N}, \tag*{(P)} \end{equation} where $\lambda \in (0, \infty)$ is a parameter and $p(x), q(x): \mathbb{R}^{N} \to \mathbb{R}$ are radial continuous functions satisfying $1 < p(x) < N$ and $p(x) < < q(x) < < p^{*}(x)= \frac{p(x)N}{N-p(x)}$.

#### Article information

**Source**

Differential Integral Equations, Volume 23, Number 1/2 (2010), 113-123.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356019390

**Mathematical Reviews number (MathSciNet)**

MR2588805

**Zentralblatt MATH identifier**

1240.35182

**Subjects**

Primary: 35A15: Variational methods 35H30: Quasi-elliptic equations 35B33: Critical exponents

#### Citation

Alves, Claudianor O. Existence of radial solutions for a class of $p(x)$-Laplacian equations with critical growth. Differential Integral Equations 23 (2010), no. 1/2, 113--123. https://projecteuclid.org/euclid.die/1356019390