Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 120, pp. 1-14.

Title: Solutions of p(x)-Laplacian equations with critical exponent and
       perturbations in R^N
  
Authors: Xia Zhang    (Harbin Inst. of Technology, Harbin, China)
         Yongqiang Fu (Harbin Inst. of Technology, Harbin, China)

Abstract:
 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.
 
Submitted June 19, 2012. Published July 19, 2012.
Math Subject Classifications: 35J60, 46E35.
Key Words: Variable exponent Sobolev space; critical exponent; weak solution.