Research On The Existence Of Solutions Of Several Types Of Nonlinear Partial Differential Equations

In this paper,the existence of solutions for two kinds of nonlinear elliptic par-tial differential equations is studied by using variational methods.The first kind is Kirchhoff equation which originates from elasticity and population dynamics.It was proposed by Kirchhof in the process of studying the classical D’Alembert wave equation of free vibration of telescopic rope.Since P.L.Lions introduced an abstract functional analysis framework,the above-mentioned problems have attracted more and more attention.The second kind is the Schrodinger-Poisson equation.Schrodinger-Poisson equation has a wide range of applications in the field of physics.In the Abelian Gauge theories,it provides a model to describe the interaction of a nonlinear Schrodinger field with an electromagnetic field,and describes the interaction between a charge particle interacting with the electromag-netic field in quantum electrodynamics.In addition,Schrodinger-Poisson equation also appears in nonlinear optics,semiconductor theory and plasma physics,and has a strong physical background.According to the content,this paper is divided into the following four chapters:The first chapter summarizes the research background and present situation of this paper and then briefly introduces our main work.In chapter 2,we mainly study a class of Kirchhoff equations with 3-superlinear growth condition where a,b,V are positive constants and 2<p<3.Using the scaling technique,we overcome the difficulty to verify the Cerami condition caused by the combination of the nonlocal term and the pure power nonlinearity,and establish the existence of infinitely many high energy solutions with the help of fountain theorem.In chapter 3,we consider the Schrodinger-Poisson equation with critical and subcritical growth conditions where V(x),K(x)and a(x)satisfy some mild conditions.Firstly,by using a direct method,we prove the existence of the ground state solution of the equation and obtain the range of ground state energy.Secondly,by employing the constraint variational method and the quantitative deformation lemma,an existence result for ground state sign-changing solution is obtained under suitable conditions on f.We also prove that the ground state sign-changing solution to this problem only changes the sign once and possesses an energy exceeding twice the least energy.The fourth chapter is the summary and prospect of this paper,and points out the related problems that can be further studied.