Positive And Negative Solutions Of A Class Of Superlinear Schrödinger Equation And Change Solution

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 of important research directions in modern mathematics. Because it can well explain various natural phenomenon, so, the mathematical world and the natural science world pay more attention to the nonlinear func-tional analysis. They have obtained some new results for mathematical physics, mathematical biology and physics, from which the nonlinear Schrodinger equa-tion stems. It becomes a very important domain of differential equation research at present, and the existence of signed and sign-changing solutions for this kind of equation is also the hot spot which has been discussed in recent years.In this paper, we use the invariant sets of descending flow, critical point theory as well as minimax methods to study the existence of signed and sign-changing solutions for superlinear Schrodinger equation (Pλ).The thesis is divided into three chapters according to contents.In chapter 1, it is concerned with the existence of sign-changing solutions for this kind of equation (Pλ). We have weaken the condition that f(x,u) is continuous, and generalize the equation we now study. Applying the invariant set method and the Principle of Symmetric Criticality, under some assumptions on a and f, we obtain an unbounded sequence of radial sign-changing solutions for the equation (Pλ) in (RN) whenλ> 0 large enough. As N= 4 or N≥6,λ> 0 given, using Fountain Theorem and the Principle of Symmetric Criticality, we prove that there exists an unbounded sequence of non-radial sign-changing solutions for the equation (Pλ) in(RN).In chapter 2, under the similar conditions of chapter 1, we consider the exis-tence of sign-changing solutions for this kind of equation (Pλ) with constructing cones different from chapter 1. Using the invariant sets of descending flow and critical point theory, we obtain an unbounded sequence of sign-changing solu-tions for (Pλ). For every x∈RN,(?) is nondecreasing on R, we can estimate the number of nodal domains of every sign-changing critical point, under some assumptions, uk has at most k+1 nodal domains.In chapter 3, using the invariant sets of descending flow as well as critical point theory and combining with corresponding eigenvalue problem, we consider the existence of signed solutions for this kind of equation (Pλ), and obtain a positive and a negative solution for (Pλ) whenλ> 0 large enough.The innovation of this paper is as follows:This paper has the advantage that we do not need f(x,u) to be continuous as in most papers on signed and sign-changing solutions for superlinear Schrodinger equation (Pλ), and we investigate more ordinary equation in this paper.