Existence Of Multiple Solutions For A Type Of Variable Exponent Problems With Non-smooth Potential

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.