The Existence Of A Class Of Nonlinear Schrodinger Equations

Along with science's and technology's development, various nonlinear prob-lem has aroused people's widespread interest day by day, and so the nonlinear analysis has become one important research directions in modern mathematics. The nonlinear functional analysis is an imporant branch in nonlinear analysis, because it can well explain various the natural phenomenon, so, the mathemat-ical world and the natural science world pay more attention to the nonlinear functional analysis.The nonlinear Schrodinger equation stems from the applied mathematics,the physics and each kind of application discipline.It is one of most active domains of the integral differential equation studiesin at present.and the existence of solutions for this kind of equation is also the hot spot at present.In this paper, we use variational methods, critical point theory,minimax methods,the mountain pass theorem, the fountain theorem as well as the linking theorem to study the existence of nontrivial solutions for a kind of nonlinear Schrodinger equation.The thesis is divided into three chapters according to contents.In chapter 1, we use variational methods and critical point theory to inves-tigate the existence of nontrivial solutions for the superlinear problems without Ambrosetti and Rabinowitz growth condition whereΩ(?) RN(N> 2)is a bounded smooth domain,f(x, u)is a continuous func-tion onΩ×R, a(x)is a non-negative continuous function onΩ.We generalize and improve the results in [6],the method is also different from [6].In chapter 2, we use the fountain theorem and minimax methods to inves-tigate the existence of multiple solutions for a class of superlinear Schrodinger equation. where a∈C(RN, R),g∈C(RN×R, R),which imposed some known results. In chapter 3,we use the linking theorems and variational methods to inves-tigate the existence of nontrivial solutions for a class of asymptotically linear Schrodinger equation. where a∈C(RN, R), f∈C(RN×R, R),which imposed some known results.First,the equation and the nonlinear condition have been improved,we inves-tigate the existence of nontrivial solutions for the superlinear problems without Ambrosetti and Rabinowitz growth condition and the methods are also different from those in[6].At last,we investigate the existence of nontrivial solutions for a class of nonlinear Schrodinger equations.The discussion is more wide and the results are better.