Applications Of Variational Methods To Quasilinear Schrodinger Equations

In this paper we consider the quasilinear Schrodinger equation Under different conditions on V and g, we obtain the existence of multiple solutions and a nontrivial solution using the variational methods. This paper is organized as follows.In the first chapter, we outline the background and introduce the structure of the dissertation and some preliminaries.In chapter2, we assume that g and V are periodic in x1,...,xN and g is odd,"super-quadratic", subcritical and satisfies a monotonicity condition in u. We employ the generalized Nehari manifold method to obtain infinitely many geometrically distinct solutions.Chapter3deals with the "asymptotically quadratic" case. We use the modified version of the generalized Nehari manifold method to get infinitely many geometrically distinct solutions. Roughly speaking, we show that there is a homeomorphic mapping between the Nehari manifold and some open subset in the unit sphere. Then we derive the deformation lemma from the discreteness of Palais-Smale sequences. Finally, we get infinite solutions by the genus theory.In chapter4, under a weak one-sided asymptotic estimate for V and the nonlinearity term at infinity, we obtain the existence of a ground state solution. In this case, the usual varia-tional methods cannot be applied in a standard way for the quasilinear Schrodinger equation. To overcome it, we first prove that the corresponding Nehari manifold is a C1manifold, and then get a Palais-Smale sequence of the functional. Finally we obtain the representation of the Palais-Smale sequence and get a ground state solution.Chapter5is concerned with the existence of a nontrivial solution for a quasilinear Schrodinger equation with sign-changing potential.Chapter6deals with the asymptotically linear Schrodinger equations. We use the gener-alized Nehari manifold method to obtain the existence of a ground state solution and infinitely many geometrically distinct solutions.