Existence Results for the p(x)-Laplacian with Nonlinear Boundary Condition

Zhiqiang Wei,Zigao Chen

doi:10.5402/2012/727398

Abstract

By using the variational method, under appropriate assumptions on the perturbation terms such that the associated functional satisfies the global minimizer condition and the fountain theorem, respectively, the existence and multiple results for the -Laplacian with nonlinear boundary condition in bounded domain Ω were studied. The discussion is based on variable exponent Lebesgue and Sobolev spaces.

Highlights

In recent years, increasing attention has been paid to the study of differential and partial differential equations involving variable exponent conditions
The interest in studying such problems was stimulated by their applications in elastic mechanics, fluid dynamics, or calculus of variations
For more information on modeling physical phenomena by equations involving p x -growth condition we refer to 1–3. The appearance of such physical models was facilitated by the development of variable exponent Lebesgue and Sobolev spaces, Lp x and W1, p x, where p x is a real-valued function

Summary

Zhiqiang Wei and Zigao Chen

Department of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450000, China Copyright q 2012 Z. Wei and Z. Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. By using the variational method, under appropriate assumptions on the perturbation terms f x, u , g x, u such that the associated functional satisfies the global minimizer condition and the fountain theorem, respectively, the existence and multiple results for the p x -Laplacian with nonlinear boundary condition in bounded domain Ω were studied. The discussion is based on variable exponent Lebesgue and Sobolev spaces.

Introduction

ISRN Applied Mathematics