Existence results for Schr\"odinger $p(x)$-Laplace equations involving critical growth in $\mathbb{R}^N$

TL;DR

This paper proves the existence of solutions for Schrödinger $p(x)$-Laplace equations with critical growth nonlinearities in $\

Contribution

It introduces new existence results for variable exponent Schrödinger equations with critical growth, utilizing concentration-compactness in weighted Sobolev spaces.

Findings

Existence of solutions under various potential conditions.

Application of concentration-compactness principles in variable exponent spaces.

Handling of critical growth nonlinearities on subsets of $\

Abstract

We establish some existence results for Schr\"odinger $p(x)$-Laplace equations in $\mathbb{R}^N$ with various potentials and critical growth of nonlinearity that may occur on some nonempty set, although not necessarily the whole space $\mathbb{R}^N$. The proofs are mainly based on concentration-compactness principles in a suitable weighted variable exponent Sobolev space and its imbeddings.