Existence Of Solutions Of The Schrödinger Equation

Along with science's and technology's development, various non-linear prob-lem has aroused people's widespread interest day by day, because it can well explain various the natural phenomenon. So, the mathemaatical world and the natural science world attach importance to the nonlinear functional analysis. They have obtained some new results for the nonlinear functional analysis and its applications. The existence of solutions as well as multiplicity, geometric property for a class of Schrodinger equation is also the hot spot which has been discussed in recent years.In this paper, under the more general superlinear condition, we use the local linking theorem, the mountain pass theory, the minimax theorems to study the existence of solutions for several kinds of Schrodinger equation and we apply the main results to the boundary value problem for the Schrodinger equation.The thesis is divided into three chapters according to contents.In chapter 1, under a weaker superlinear assumption, we study the existence of a nontrivial solution for a class of superlinear elliptic problems by the local linking theorem theorem. where a∈LP(Ω), P>N/2,g∈C(Ω×R,R),Ω(?) RN(N≥3) is a bounded domain whose boundary is a smooth manifold. Our results generalize many recent studies.In chapter 2, by using the generalized mountain pass theorem and approach-ing method, we consider the existence of a nontrivial solution for nonlinear Dirich-let problem. Then it is considered continuous dependence of solutions for pa-rameter, where a∈L∞(Ω),Ω(?) RN(N≥3) is a bounded domain whose boundary is a smooth manifold, g(x, u) is a continuous function onΩ×R. Our results generalize some known results. In chapter 3, by using the critical point theorem under the weak topology, it is concerned with the existence of a nontrivial solution as well as multiplicity results and continuous dependence of solutions for parameter for semilinear Schrodinger equation. whereλ> 0, most of papers consider the periodic or the radially symmetric problems. Here we deal with non-periodic and non-radially symmetric problems.