Ehari Manifold And The Existence Of The Positive Solutions For The Nonlinear Elliptic Equations

In this paper, we investigate the existence of the positive solutions for the nonlinear elliptic equations, which is based on the establishment of the Nehari manifold.In chapter one, that is the introduction part. Firstly, we introduce the back ground of the development of the nonlinear partial differential equations, its vital role and important position; Secondly, we introduce the development course of Variational Methods and its research progress; Finally, the chapter briefly explain the generation and application of the Nehari manifold.In chapter two, we mainly study the existence of the multiple positive weak solutions for the Dirichlet problem of the nonlinear partial differential equations with singularity. Basing on the extraction of Palais-Smale sequences in the Nehari manifold, we exploit the relationship between the Nehari manifold and fibering maps, and combing the two compact proposition. We get the equation have two positive solutions.In chapter three, we discuss the multiplicity of positive solution to quasilinear elliptic equations involving the critical situation for the Neumann problem.By splitting Nehari manifold Nλ into three parts:Nλ+, Nλ-and Nλ0, We give a variational method, by using it, we do this without the extraction of Palais-Smale sequences in the Nehari manifold.