Existence Of Soultions For Several Nonlinear Elliptic Equations

In this work,we study the existence of solutions for three classes of nonlinear elliptic equations.The paper is organized into five chapters and the main content proceeds as follows:Chapter 1 serves as an introduction,providing a brief background on nonlinear elliptic equations,along with examples illustrating the research status of these equations in the literature.The chapter concludes by presenting the basic definitions and theorems required to solve the problems discussed in this paper.In chapter 2,we using variational methods,combining with the symmetric mountain pass lemma and the（C）_c condition to prove that the existence of solutions for a class of pKirchhoff type elliptic equation M（∫_R^N（|▽u|^p+V（x）|u|^p）dx）(-Δ_pu+V（x）|u|^p-2u)=f（x,u）,x∈R^N,where the function V and nonlinear termf satisfy certain conditions,M（t）=t^k,k>0.In chapter 3,we using the mountain pass lemma and（PS）_c condition to prove that the existence of weak solutions to the following nonlinear fractional Schrodinger equation（-Δ）^su+V（x）u=f（x,u）,x∈R^N,where 0<s<1,the potential functions V（x）and nonlinear term f satisfy certain conditions.In chapter 4,by using the Nehari manifold and a fibering map,we consider the existence of positive solutions for a class of quasilinear elliptic equation boundary value problem （?） where 1<q<p<r<p^*,Ω（?）R^N is a bounded domain with smooth boundary.In chapter 5,we have a summary of the content of the first four chapters and look forward to future research work.