Local existence conditions for an equations involving the $p(x)$-Laplacian with critical exponent in ${\mathbb{R}}^N$

Abstract

The purpose of this paper is to formulate sufficient existence conditions for a critical equation involving the $p(x)$-Laplacian posed in ${\mathbb{R}}^N$. This equation is critical in the sense that the source term has the form $K(x)|u{|}^{q(x)-2}u$ with an exponent $q$ that can be equal to the critical exponent $p^{\ast }$ at some points of ${\mathbb{R}}^N$ including at infinity. The sufficient existence conditions we find are local in the sense that they depend only on the behaviour of the exponents $p$ and $q$ near these points. We stress that we do not assume any symmetry or periodicity of the coefficients of the equation and that $K$ is not required to vanish in some sense at infinity like in most existing results. The proof of these local existence conditions is based on a notion of localized best Sobolev constant at infinity and a refined concentration-compactness at infinity.