Existence of solutions to the Steklov $p(x)$-Laplacian problem involving the critical exponent

Abstract

In this paper, we consider a class of Steklov $p(x)$-Laplacian problems with a critical exponent, given by the following equation:
$$
\left\{
\begin{array}{ll}
(-\Delta)_{p(x)}u=|u|^{r(x)-2}u+ f(x,u),
\quad \mbox{in }\Omega, \\
\\
|\nabla u|^{p(x)-2}\frac{\partial u}{\partial v}= |u|^{s(x)-2}u, \quad \mbox{on }\partial\Omega,
\end{array}
\right.
$$
where $\Omega \subset \mathbb{R}^N (N \geq 2)$ is a bounded domain with Lipschitz boundary $\partial\Omega$. Here, $\frac{\partial}{\partial v}$ denotes the outer unit normal derivative, he function $f: \Omega \times \mathbb{R} \rightarrow \mathbb{R}$ is a Carathéodory function satisfying appropriate assumptions, and the functions $p$ and $r$ are continuous in $\overline{\Omega}$, such that $1 < r(x) \leq p^{*}(x)$ for all $x \in \Omega$, where $p^{*}(x)$ represents the critical Sobolev exponent.

To establish the existence and multiplicity of solutions, we employ variational methods, including the mountain pass theorem and symmetric mountain pass theorem, combined with the concentration-compactness principle. These techniques enable us to find solutions that satisfy the given boundary conditions and exhibit interesting properties related to the critical exponent.