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

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.