Research On Existence Of Solutions For Two Classes Of Nonlinear Elliptic Equations

In this paper,the existence problems of solutions for two kinds of nonlinear elliptic equations are studied.The first class is the p-Laplace quasilinear elliptic equations with Hardy-Sobolev critical exponents.By establishing the embedding theorem in Sobolev space,we directly obtain the strongly converging sequence in Sobolev space,and obtain the existence of radial and non-radial solutions by using the minimization method.The second class is Kirchhoff type equations with singular nonlinear terms on compact Riemannian manifolds.We obtain the existence and uniqueness of the solutions by Nehari manifolds and Ekeland variational principle.The main structure of this paper is as follows:In Chapter 1,the research background and development status of nonlinear elliptic equations with critical exponentials and Kirchhoff equations are described,and the main work and research characteristics of this paper are briefly described.In Chapter 2,this paper mainly introduces the basic knowledge of definition and related theorems used in this paper.In Chapter 3,the existence of radial and non-radial solutions for p-Laplace quasilinear elliptic equations with Hardy-Sobolev critical exponentials is studied.Firstly,the compact embedding theorem from Sobolev space to weighted Lebesgue space is obtained based on Lions lemma and nonlinear functional theory.Secondly,Pohozaev identity is used to prove that there is no nontrivial solution in the star region.Finally,according to the compact embedding theorem and minimization technique,the p-Laplace quasilinear elliptic equation with Hardy-Sobolev critical exponentials has radial solutions in the whole space,and radial and non-radial solutions in the bounded region.In Chapter 4,the existence and uniqueness of solutions of strongly singular Kirchhoff type equations on closed Riemannian manifolds are studied.Firstly,two constraint sets are obtained by fiber mapping and Nehari manifold,and the corresponding properties of these two constraint sets are obtained.Secondly,the existence and uniqueness of solutions of strongly singular Kirchhoff type equations on closed Riemannian manifolds is obtained by Ekeland variational method.In Chapter 5,the research content of this paper is summarized,and the further thinking and prospect of the research of these two kinds of nonlinear elliptic equations are put forward.