Solvability Of Nonlinear Singular Or Degenerate Elliptic Equations (Systems)

In this thesis, via the Leray-Lions theorem and some fixed point theorems, we discuss solvability of a class of nonlinear singular or degenerate elliptic equations and systems.This thesis consists of four pats. Firstly, the solvability of the boundary value problems for degenerate quasilinear elliptic equations of higher order is discussed . Secondly, the existence and multiplicity of positive radial solutions of the boundary value problem for a class of quasilinear elliptic equations is proved. Then, existence results of entire radial solutions for a class of singular polyharmonic equations are given. At last we study the existence of positive radial solutions to quasilinear elliptic equations.In chapter one , different from most of papers, the problem is discussed in the weighted Sobolev spaces. Via the Leray-Lions theorem, we present a general existence theorem for a class of degenerate elliptic boundary value problems. In our discussion, the Leray-Lions condition is weakened.In chapter two , using the fixed point theorem for cone expansion/compression we obtain the existence of at least three positive radial solutions for a class of quasilinear elliptic equations in a ball. When the nonlinear reaction term f(t, s) is increasing on s the existence of positive radial solution is also obtained via the technique of iteration.In chapter three, the existence and multiplicity results on the positive entire solutions of a class of singular nonlinear polyharmonic equations in R~n are established. The main tool in the proof is the Schauder-Tychonov fixed point theorem.Chapter four, using the contraction mapping principle and the shooting method, discusses the existence and uniqueness of the local solutions and existence of the global solutions to a class of quasilinear degenerate elliptic systems with the p-Laplace-like operator as its principal and with singularity on the boundary.