Electron. J. Diff. Equ., Vol. 2012 (2012), No. 120, pp. 1-14.

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

Xia Zhang, Yongqiang Fu

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.

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Xia Zhang
Department of Mathematics, Harbin Institute of Technology
Harbin 150001, China.
Department of Mathematics, Pohang University of Science and Technology
Pohang, Korea
email: piecesummer1984@163.com
Yongqiang Fu
Department of Mathematics, Harbin Institute of Technology
Harbin 150001, China.
email: fuyqhagd@yahoo.cn

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