Existence of solution to a critical equation with variable exponent

In this paper we study the existence problem for the p(x)-Laplacian operator with a nonlinear critical source. We find a local condition on the exponents ensuring the existence of a nontrivial solution that shows that the Pohozaev obstruction does not holds in general in the variable exponent setting. The proof relies on the Concentration-Compactness Principle for variable exponents and the Mountain Pass Theorem.