The Existence And Multiplicity Of Solutions For The Schrodinger-Maxwell System

In this doctoral dissertation, by using by using variational methods and critical point theory, we study the existence and multiplicity of the solutions for Schrodinger-Maxwell systems. This dissertation consists of fiveIn Chapter one, we sketch the historical background, status, the up-to-date progress for the discussed problems, our main work, some preliminaries of variational methods and critical point theory.In Chapter two, by using symmetric mountain pass theorem, we shall be con-cerned with the study of the existence of infinitely many solutions for the nonlinear Schrodinger-Maxwell systems We removed two basic requirements in the existed literature:(i) V(x) is positive definite;(ii) limf(x,t)/t=0uniformly for x∈R3. Some weaker sufficient conditions are obtained in the case when V(x) and tf(x, t) are allowed to be sign-changing. And our results improve and generalize the existing results.In Chapter three, we study the existence of the Nehari-type ground state solu-tions for the Schrodinger-Maxwell system In this chapter, we develop the Nehari techniques. By using this techniques, Some weaker sufficient conditions are obtained in the case when V(x), K(x) and f(x, t) are periodic or asymptotically periodic in x. So far as we know, no results were obtained in the existed literatures.In Chapter four, by using mountain pass theorem, we study the existence of nontrivial solutions for the periodic and asymptotically cubic nonlinear Schrodinger-Maxwell systems Some weaker sufficient conditions are obtained. So far as we know, no results were obtained in the existed literatures. In Chapter five, we consider the existence of semiclassical solutions for the Schrodinger-Maxwell systems By constructing a simple test function, we give out a definite upper bound of the critical level value, by using the mountain pass theorem, we search the Cerami se-quence below the critical level value. Finally, the required semiclassical solution was obtained. Compared to the existed literature, our approach is more straightforward, our conditions are weaker. In particular, we give an upper bound of the small per-turbed parameter ε.