Solutions of p(x)-Laplacian equations with critical exponent and perturbations in R^N

Based on the theory of variable exponent Sobolev spaces, we study a class of $p(x)$-Laplacian equations in $mathbb{R}^{N}$ involving the critical exponent. Firstly, we modify the principle of concentration compactness in $W^{1,p(x)}(mathbb{R}^{N})$ and obtain a new type of Sobolev inequalities involving the atoms. Then, by using variational method, we obtain the existence of weak solutions when the perturbation is small enough