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

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.