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 $p(x) < < q(x) < < p^{*}(x)= \frac{p(x)N}{N-p(x)}$.