【作者】 刘丽丽；

【导师】 葛斌；

【作者基本信息】 哈尔滨工程大学 ， 应用数学， 2017， 硕士

【摘要】 变指数问题来源于电子流变流体学和非线性弹性力学的研究,特别是对非标准增长的椭圆微分方程的研究,在近些年来得到了国内外许多学者的广泛关注。其主要原因在于该问题有重要的应用背景,诸如,图像恢复、稳定热变粘性流和多孔介质气体渗透。近十年来,对有变指数椭圆型偏方程多解性的研究已成为国际热点问题,许多研究成果针对的是具有光滑位势的偏微分方程,然而,对非光滑位势对应的微分包含方程研究尚少。基于此,本文在全空间R~N中,讨论如下类型的具有变指数p(x)-Laplacian微分包含问题:本文包括以下四个部分:第一章,简述了论文的研究背景、国内外研究现状以及本文的主要研究工作。第二章,主要介绍了本文所用到的变指数问题的相关理论,即Lebesgue空间、Sobolev空间、临界点理论、广义导数以及亏格理论。第三章,我们讨论了针对本文所提到的一类具有非光滑位势的变指数拉普拉斯问题非平凡解的存在情况,给出了定理1.1,并利用非光滑山路定理加以证明。第四章,在对称条件F(x,-t)=F(x,t)下,本文通过亏格理论,讨论了问题(P)无穷解的存在性(即定理1.2)。更多还原

【Abstract】 Variable exponent problem originates nonlinear elastomechanics and electrorheological fluids.Recently,because of important applications,variable exponent problem have gained widely attention of the people,especially,nonlinear elliptic partial differential equations.such as image processing,the mathematical description of the processes filtration of an idea barotropic gas through a porous medium and stationary thermorheological viscous flows.In the recent 10 years,nonlinear elliptic partial differential equations with variable exponent have been many research works on this topic.Most of them are restricted to the elliptic PDE’s with smooth potential.However,studying the differential inclusion problems involving with non-smooth potential is seldom.So,the thesis studies a variable exponent p(x)-Laplacian problems on R~N.This article includues the following four parts:In chapter 1,we present the research background,the domestic and international research status and the main research work in this paper respectively.In chapter 2,we investigate the variable exponent problem,Lebesgue space,Sobolev space and the critical point,generalized gradients variational method and the genus.In chapter 3,we disscuss the existence of multiple solutions for a types of variable exponent Laplacian problems with the non-smooth potential,and establish theorem 1.1 by the mountain theorem.In chapter 4,we study on existence multiple solutions(theorem 1.2)for a types of variable exponent Laplacian problems with the non-smooth potential,under suitable oscillatory assumptions on F(x,-t)=F(x,t)by genus.更多还原