A Few Kinds Of Schrodinger Equation Of The Existence Of Nontrivial Solutions

With the development of science and technology, the nonlinear functional analysishas become one important research directions in modern mathematics both to have theprofound theory and to have the widespread application. It takes the nonlinear prob-lems appearing in mathematics and physics as background to establish some generaltheories and methods to handle nonlinear problem. The nonlinear diferential equationof nonlinear stems from the applied mathematics, the cybernetics, the physics and eachkind of application discipline. It is an important kind of question in the diferentialequations, it is one of most attention domains of functional analysis studies at present,so it attaches more and more attention.The nonlinear Schro¨dinger equation stems from the applied mathematics, thephysics and each kind of application discipline. It is one of most active domains ofthe integral diferential equation studies in at present and the existence of solutions forthe kind of equation is also the hot spot at present. Many researchers using mountainpass lemma, a weak topology and the linking theorem, they obtained the solution ofthe Schr¨odinger equation.Research of Kirchhof type problem has developed rapidly in recent years. Kirch-hof type problems is related to the stationary analogue of the equation which proposedby Kirchhof [16] as an extension of the classical D’Alembert wave equation for freevibrations of elastic strings. So far there have been many authors to study this prob-lem, the main methods have the invariant sets of descent flow, Yang index, mountainpass lemma, Morse theory and the local linking. Furthermore, X.Wu considered theSchro¨dinger-Kirchhof type problems.In this paper, we use strong sandwich pairs, Mountain pass lemma and criticalpoint theory, to research the existence of solutions for the Schro¨dinger equation withasymptotically linear and Schro¨dinger-Kirchhof type problems.The thesis is divided into three sections according to contents.Chapter1We introduce some fundamental knowledge and source of theory.Chapter2We consider the existence of solution to the Schr¨odinger equationswith asymptotically linear nonlinearities V can be sign-changing. Also, the functional is not semibounded and the separate functional vanished. We study the existence of nontrivial solutions for problem (2.1.1) in a concrete condition, using strong sandwich pairs and variational methods.Chapter3The following Schrodinger-Kirchhoff type problems without the Palais-Smale condition is well discussed, where constants a, b>0, N=1,2, or3, and f∈C(RN×R,R). Also, with satisfy weaker assumptions than those used in the literatures, nontrivial solutions and a se-quence of high energy solutions are established, using the mountain pass lemma and the symmetric mountain pass lemma.