$\mathbf{R}^{\bm N}$\,上的\,${\bm p}({\bm x})$-Laplace问题的多解性

华东师范大学学报(自然科学版) ›› 2012, Vol. 2012 ›› Issue (5): 109-119.

$\mathbf{R}^{\bm N}$\,上的\,${\bm p}({\bm x})$-Laplace问题的多解性

陈自高

华北水利水电学院~~数学与信息科学学院, 河南~~郑州 450011

收稿日期:2011-12-01 修回日期:2012-03-01 出版日期:2012-09-25 发布日期:2012-09-29

Multiple solutions for ${\bm p}({\bm x})$-Laplacian problems in ${\bf R}^{\bm N}$

CHEN Zi-gao

Department of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou Henan} 450011, China

Received:2011-12-01 Revised:2012-03-01 Online:2012-09-25 Published:2012-09-29

摘要/Abstract

摘要： 在扰动项\,$f_1(x,u),\, f_2(x,u)$~中, 其中一项是超线性并且满足\,Ambrosetti-Rabinowitz\,条件, 另一项为次线性的情形下, 分别利用``喷泉定理''和``对偶喷泉定理'' 研究了无界区域\,$\mathbf{R}^{N}$\,上的\,$p(x)$-Laplace\,方程解的存在性和多解性问题. 此问题是基于变指数\,Lebesgue\,和\,Sobolev\,空间进行讨论的.

关键词: 变指数\,Sobolev\,空间, $p(x)$-Laplacian, (PS)$_c^\ast$条件, 喷泉定理, 对偶喷泉定理

Abstract: By using the fountain theorem and the dual fountain theorem, respectively, the existence and multiplicity of solutions for $p(x$)-Laplacian equations in $\mathbf{R}^{N}$ were studied, assumed that one of the perturbation terms $f_1(x,u),\, f_2(x,u)$ is superlinear and satisfies the Ambrosetti-Rabinowitz type condition and the other one is sublinear. The discussion was based on variable exponent Lebesgue and Sobolev spaces.

Key words: variable exponent Sobolev spaces, $p(x)$-Laplacian, (PS)$_c^\ast$ condition, fountain theorem, dual fountain theorem

中图分类号:

O175.2

陈自高. $\mathbf{R}^{\bm N}$\,上的\,${\bm p}({\bm x})$-Laplace问题的多解性[J]. 华东师范大学学报(自然科学版), 2012, 2012(5): 109-119.

CHEN Zi-gao. Multiple solutions for ${\bm p}({\bm x})$-Laplacian problems in ${\bf R}^{\bm N}$[J]. Journal of East China Normal University(Natural Sc, 2012, 2012(5): 109-119.