Existence And Multiplicity Of Solutions For Elliptic Problems With Electromagnetic Fields And Critical Nonlinearity

In this paper we consider some p-Laplacian elliptic problems with electromagnetic fields and critical nonlinearity. Under different non-linear terms and regions, we obtain the existence of nontrivial solution and multiple solutions by using the variational methods. This paper is organized as follows.In the first chapter, we outline the background and significance, introduce the structure of the dissertation and some preliminaries.In chapter 2, we dealt with the p-Laplacian equation with electro-magnetic fields and critical nonlinearity, where potential function V(x) has critical frequency. Due to the lack of compactness conditions caused by critical nonlinear term, we will use the concentration compactness principle to overcome the difficulties, and the method in this paper is very different with previous literatures [35,36]. More importantly, our method in this paper is also suitable for other potential functions. In the end, we obtain the existence of nontrivial solutions and multiple solutions by using the variational method.Chapter 3 consider p-Laplacian type elliptic problems with elec-tromagnetic fields and critical nonlinearity, here Ω(?)RN is an open bounded domain with smooth boundary. On the one hand, due to a lack of compactness conditions caused by critical nonlinear term, we will use the concentration compactness principle to overcome the difficulties. On the other hand, caused by the energy functional neither upper nor lower bound, we will use the truncation method to solve this difficulty. We prove the existence of infinitely many solutions of this problem which tend to zero by using a new version of the symmetric mountain-pass lemma due to Kajikiya [51].In chapter 4, we consider Kirchhoff type problems with electro-magnetic fields and the critical growth in RN. We can prove that compactness conditions by choosing appropriate parameters α and β, together with concentrated compactness principle. We can also prove the existence of infinitely many solutions of this problem which tend to zero by using a new version of the symmetric mountain-pass lemma due to Kajikiya [51].