Existence,Uniqueness And Multiplicity Of Solutions For A Class Of Quasilinear Equations With Singularity

Quasi-linear singular equations play an important role in many physical fields.For example,it is a supernuid film equation in plasma physics;it describes the self-channel effect of ultrashort lasers,which has attracted the attention of many authors.This paper will use variable substitution method and Schauder fixed point theorem to study the existence and uniqueness of positive solution of quasilinear equation with singularity,and the perturbation method is used to obtain the multiplicity of positive solutions of quasilinear singular equation under subcritical condition.The thesis consists of three sections.Chapter 1 is the preface.In Chapter 2,we mainly discuss the existence and uniqueness of positive solution con-cerning the following equation:where ?(?)RN(N?3)is a bounded area with smooth boundaries.The main methods we use are perturbation methods,Schauder fixed point theorems,and variable substitution methods.We get the following main conclusions:Assuming ??(0,1),then the above-mentioned equation has a unique solution u? C2(?)?H01(?).The third chapter discusses the multiplicity of solutions about the following singular quasilinear equation:where ?(?)RN(N?3)is a bounded area with smooth boundaries,and 0? ?,2*=2N/(N-2),? ?(0,1),? is a positive parameter,?4u=div(|?V|2?u).The functional is nondifferentiable as a result of the singular term.Thus it would be well if we adopt perturbation method in this chapter.Firstly we discuss the multiplicity of the solutions to the perturbation problem,and then take the limit for disturbance parameter and get the solutions to the original equation.We get the main conclusion:Assuming ??(0,1),then there exists ?*>0,such that for any ??(0,?*),above question has two different solutions in W01,4(?).