A Study Of A Class Of Elliptic Equation And System With Nonlinear Boundary Conditions

This paper mainly deals with some elliptic equation and elliptic system with nonlinear boundary conditions. Some results of the existence of solutions are demonstrated by employ the blow-up technique,prior estimate and Moser iterative method .A series of new results are obtained in the literature. the whole paper is arranged as follow:As an introduction ,in Chapter 1,the background and obtained results are briefly addressed. At the same time, we introduce the main results and the method we used in this thesis.In chapter 2, we collect some basic materials in partial differential equation and some important theorems which will be used in the later chapter such as Soblev space, Maximum theorem and Schauder estimates etc.In chapter 3, we prove the existence of general solution for the equation with nonlinear boundary condition via Mountain pass lemma, and then, Moser iterative method is used to improve the regularity of the general solution, finally, we obtain the proof of existence of classic solution.In chapter 4, we prove the existence of nontrivial positive solution of the system with nonlinear coupling through the boundary conditions under suitable assumptions, In the proof we use a fixed point argument and the key ingredient is a Liouville type theorem for a system of Laplace equations with nonlinear boundary conditions on the half space.