The Application Of The Calculus Of Variations In Several Classes Of Schrodinger Equations

The Calculus of Variations originates from various of parts of applied mathe-matics and physics. It is one of the more active fields that are studied in nonlinear functional analysis. The systems of Critical Point are important parts in The Cal-culus of Variations. The structures they present have the profound significance of physical backgrounds and mathematical models. So, the study of schrodinger equation and furthermore the systems of schrodinger-poisson equations has profoundly intrinsic value.They play important parts in applied mathematics and physics, especially in the Plasma Physics and the Quantum Mechanics. Therefore, it becomes of great impor-tance to study the existence and multiple solutions of them. Furthermore, the study of the nature of the solutions is also important.The present paper employs the The Calculus of Variations such as the properties minimizing sequence, variant fountain theory and so on, to investigate the existence of solutions of schrodinger equation and schrodinger-poisson equations. The obtained results are either new or intrinsically generalize and improve the previous relevant ones under weaker conditions.The thesis is divided into three sections according to contents.Chapter 1 Preference, we introduce the main contents of this paper and some important base theory.Chapter 2 In this paper we study the semi-linear schrodinger equation:-Δu+V(x)u=K(x)|u|2*-2u+g(x,u),u∈H1(RN).We use the properties minimizing sequence which assure the existence of ground state solutions. Under a Nehari type condition, we show that the standard Ambrosetti-Rabinowitz super-linear condition can be replaced by a more natural super-quadratic condition.We consider two cases of the potentials, one is periodic; the other is when V has a bounded potential well.Case one:Base on the page five of this paper's conditions, using the properties mini-mizing sequence, we get the result: Theorem 2.2.1 Under assumptions(V1),(go)一(94)(In the five page of this paper).Eq.(2.1)(In the four page of this paper) has a weak solution,u∈M,such thatΦ(u)=c>0,c is defined as c=infNΦ(u). Case two:We get the similar re-sult by using the similar calculus.Chapter 3 In this paper we study the Schrodinger-Poisson equations in the three-dimensional space: Using the variant fountain theorem introduced by zou[36],we get the result of inpinitely many large solutions for this Schrodinger-Poisson equations:Theorem 3.1.1 Under the assumptions(V1),(f1)-(f5)(In the eighteen page of this paper),andε∈[(?)], system (3.3)(In the eighteen page of this paper)has infinitely many solutions{(uk,φk)}such that when k→∞,we have...