Research On The Existence Of The Solution For Some Kinds Of Kirchhoff Type Partial Differential Equations

Kirchhoff type Equations has been put forward by Kirchhoff when he was studying the length vibration of the string. It is a typical equation in the nonlinear field. In recent years, its unstable and stable conditions has been researched widely.In this paper,Firstly the present paper is devoted to the investigation of a Kirchhoff-beam Equation with nonlinear boundary conditions by means of the Faedo-Galerkin method. Finally existence and uniqueness of a globle solution are proved. Secondly the existence of positive solutions for a kind of nonlocal Kirchhoff type equations in a smooth bounded domain of RN was investigated. The corresponding energy functional of the above problem were given in this paper. Then the desired solutions were equivalent to the critical points of the energy functional by variational methods. In order to deal with the condition of nonlocal term growth faster, truncation technology was used. When the nonlinear term met the corresponding as-sumptions, it was proved that the truncation problem had at least one nontrivial solutions via monotonicity trick. Finally, it was proved that the solution of the truncation problem is the solution of the original problem by using uniform estimates of elliptic equations which was given by Gidas and Spruck. The full text structure is as follows:The first chapter briefly introduces the development and current situation of the research, the main work in this paper, and at the same time the main results are obtained.The second chapter introduces some basic knowledge used in this paper, including the basic space as well as their relationship, lemma, concept and hypothesis, as well as some commonly used inequality.The third chapter is devoted to the initial-boundary value problem of a nonlinear Kirchhoff-beam Equation with nonlinear boundary conditions by means of the Faedo-Galerkin method, final the existence and uniqueness of a globle solution are proved.The fourth chapter by variational methods, the existence of positive solutions for a kind of nonlocal Kirchhoff type equations in a smooth bounded domain of RN was investigated.