Research On Some Classes Of Degenerate Elliptic Equations And Systems

In this dissertation,we discuss some classes of degenerate elliptic equations and sys-tems,which are semi-linear with singular nonlinear terms.They can be used to simulate various physical phenomena and are widely used in many fields such as fluid mechanics and biology.The study on the properties of the solutions of these equations and system-s is a hot topic in recent years.New Sobolev type embedding theorems are established in the weighted Banach space.Then some properties of the non-negative solutions and the corresponding Liouville-type theorems of the degenerate elliptic equations and systems are studied.Finally,the related properties of the nonnegative solutions of the degener-ate elliptic systems with radial arid non-radial symmetry and corresponding Liouville-type theorems are discussed.The main part of dissertation is in six chapters.In Chapter 1,we first introduce the research background and development status of semi-linear elliptic equations and systems with singular nonlinearities.Then we explain the structure and main conclusions of this dissertation.Finally,we briefly review some basic knowledge of nonlinear analysis and elliptic equations used in the following chapters.In Chapter 2,we establish new Sobolev type embedding theorems for the weighted Banach space in bounded open domain containing the origin in the whole space and obtain a set of singular positive radial symmetric entire solutions of degenerate elliptic equation in RN.Using such embeddings and the properties of the entire solutions,we construct positive weak solutions of degenerate elliptic equations with a prescribed singularIn Chapter 3,extending the weights in Chapter 2 to more general cases,we prove more general Sobolev type embedding theorems of weighted Banach space in bounded and unbounded domains,respectively..The main results give some generalizations of the well-known Caffarelli-Kohn-Nirenberg inequality.Using the established embeddings,we obtain the existence and properties of weak solutions of some related degenerate elliptic problems.In Chapter 4,by removing some crucial restrictions on the weights in the Chapter 3,we obtain new embeddings of some weighted Sobolev spaces in bounded and unbounded domains.The main results give some generalizations of the embeddings obtained in Chap-ter 3.Some applications of these embeddings to the existence of positive solutions and the regularity results of degenerate elliptic Dirichlet problem on the smooth bounded domain of the whole space are also given.In Chapter 5,we first establish Liouville-type theorems for the radial symmetric so-lution of the degenerate elliptic system on the whole space without the origin.Then we discuss properties of the positive radial symmetric solution near the origin in the unit sphere with the origin removed,and finally we get the existence of positive solutions of Dirichlet problem on unit sphere with the help of the Schauder fixed point theorem.In Chapter 6,removing the assumption that the solution is radial symmetry in Chapter 5,we use the spherical mean method and moving spherical method to obtain the Liouville type theorems of the solution of the degenerate elliptic system on the whole space without the origin.