Concentration-compactness principle for variable exponent spaces and applications

Abstract. In this paper we extend the well-known concentration – compactness principle of P.L. Lions to the variable exponent case. We also give some applications to the existence problem for the p(x)−Laplacian with critical growth. 1. Introduction. When dealing with nonlinear elliptic equations with critical growth (in the sense of the Sobolev embeddings) the concentration – compactness principle of P.L. Lions, see [12], have been proved to be a fundamental tool in order to prove existence of solutions. Just to cite a few, see [1, 2, 3, 7, 4, 11] but there is an impressive list of references on this