The Solution Of The Schrodinger Equation And Kirchhoff Type Problems

With the continuous development of science and technology, all kinds of non-linear problem have aroused people's wide attention. Nonlinear partial differential equations stem from applied mathematics, physics, control theory and other ap-plied sciences, they are the most active research topics in the field of nonlinear science, the Schrodinger equation problems and the Kirchhoff-type problems are the hot topics these years, and they are also quite important research fields of the partial differential research.The thesis is divided into third chapters.The first chapter is preliminary knowledge. Introduce some definitions and some important the theorems which can be used during proof.This chapter in-cludes second sections. In the first section, we give some definitions which can be used during proof. In the second section, we give some important inequalities and theorems which can be used during proof.In chapter 2, We consider the existence of infinitely many positive solutions for the non-periodic Schrodinger equations whereΩis a smooth bounded domain of RN and f:Ω×Râ†'R is a Caratheodory function, and 3 (?) t*>0, such that And f(x,t) satisfies the following conditions(f2) For every n∈N, there existsξn,ξ′n∈R with 0≤ξn<ξ′n and limnâ†'+∞ξ′n= 0 such that(f3) There exists a non-empty open setΩ0 (?)Ω, a constant M≥0 and a sequences{tn}n∈N (?) R+\0 with limnâ†'+∞tn=0, such that By searching for the local minima of the energy functional, We obtain a sequence of a.e. positive weak solutions of the Schrodinger equation (2.1.1).This chapter contains two sections. The first section is our introduction. We introduce the importance of Schrodinger equation problems, as well as some pre-vious research results. In the second section, by searching for the local minima of the energy functional, we prove the existence of infinitely many positive solutions for the non-periodic Schrodinger equations.In chapter 3, we study Kirchhoff type problems whereΩis a smooth bounded domain of RN and f:Ω×Râ†'R satisfies the following conditions(f1)|f(x,u)|≤C(|u|p-1+1)for some 40 such that H(x,t)≤H(x, s)+C*, for all 0