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. 
Juli\'an Fern\'andez Bonder, Nicolas Saintier and Anal\'ia Silva
math.AP