## Differential and Integral Equations

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

Claudianor O. Alves

#### Abstract

Using variational methods we establish the existence of solutions for the following class of $p(x)$-Laplacian equations: $$-div(|\nabla u|^{p(x)-2}\nabla u) + |u|^{p(x)-2}u= \lambda |u|^{q(x)-2}u + |u|^{p^{*}(x)-2}u, \,\,\,\,\, \mathbb{R}^{N}, \tag*{(P)}$$ 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

#### 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