Study On Positive Radial Solutions Of Boundary Value Problems For Elliptic Equations And Positive Solutions Of Elliptic Equations In Hole Region

There are many methods to study nonlinear elliptic partial differential equations,such as fixed point theorem,upper and lower bound method,topological degree theory and so on.In this paper,we mainly use fixed point theory to solve two kinds of problems.One is to prove the existence of positive radial solutions of boundary value problems for a class of semilinear Elliptic equations.The known problem is transformed into its radial form by radial transformation,and then the existence,uniqueness and nonexistence of its radial form solutions are discussed by using fixed point theorem,and corresponding examples are given to illustrate the theorem.In the second part,we discuss the existence and uniqueness of positive solutions of semilinear Elliptic equations in the cavity region.This part mainly transforms the problems in the bounded cavity region into the bounded smooth region which we are familiar with.When the non-linear term of the problem satisfies some specific conditions,we use the fixed point theorem to prove the existence and uniqueness of positive solutions.The existence and uniqueness of solutions to this kind of problems are given.