Local existence conditions for an equations involving the p(x) -Laplacian with critical exponent in RN

Abstract

The purpose of this paper is to formulate sufficient existence conditions for a critical equation involving the p(x)-Laplacian of the form (0.1) below posed in RN. This equation is critical in the sense that the source term has the form K(x) | u| q ( x ) - 2u with an exponent q that can be equal to the critical exponent p∗ at some points of RN including at infinity. The sufficient existence condition 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.