Research On Dirichlet Problem Of Elliptic Equation With Gradient Term In Annular Domain

What is known to us all that elliptic equation,an important branch of partial differential equation,has attracted the attention of scholars at home and abroad.The main reason why elliptic equation is focused on widely is that it is closely related to many mathematical and physical problems,for instance,incineration theory of gases,solid state physics,electrostatic field problems,variational methods and optimal control.The existence of solutions for Laplacian equation in annular domain has always played an vital role in the study of elliptic equation.A lager number of important results have been obtained in the efforts of mathematical researchers.In this paper,we study the existence of solutions for Dirichlet problem of elliptic equation with gradient term in annular domain.Consider the following problem:#12 where(?)is an annular domain in Rn,n>2,the nonlinearity(?)is continuous and C1 with respect to(u,p).This paper mainly refers to two aspects.On one hand,we want to verify the existence of solutions when nonlinearity satisfies asymptotically linear con-dition.To be precise,we first derive an ordinary differential equation which is equivalent to the above elliptic equation.Basing on Sturm-Picone comparison theorem,the existence of solutions for the equation boils down to the existence of fixed point of equivalent operator in function space,with aid of asymptoti-cally linear condition.After we prove the operator is continuous,compact and bounded in function space,we establish the existence of solutions for ordinary differential equation,in which the nonlinearity does not contain the derivative term,by Schauder fixed point theorem.And then combining with Schauder fixed point theorem again,the existence of solutions for equivalent ordinary differential equation with derivative term can be gainedOn the other hand,we investage the multiplicity of solutions when nonlin-earity satisfies superlinear conditions without classical Ambrosetti-Rabinowitz condition.We first establish the corresponding Euler-Lagrange functional And then,we know the functional satisfies the mountain pass geometry and Palais-Smale condition basing on the superlinear conditions.Finally we use iterative method to overcome the difficulties caused by the derivative term,and prove that the equation has a positive solution and a negative solutionThis paper is organized as follows.The paper is divided into five chapters The first chapter is the introduction,which introduces the background,main results and the definitions and lemmas needed in this paper.The latter four chapters are the main parts of this article.In the second chapter,we derive the elliptic equation into the equivalent ordinary differential equation.In the third chapter,we prove that the equivalent ordinary differential equation has nontrivial solution under asymptotically linear condition by using Schauder fixed point theorem.In the fourth chapter and fifth chapter,by using Mountain pass theorem and iterative method which can overcome the difficulties caused by derivative term,we prove that the equivalent ordinary differential equation has a positive solution and a negative solution under the superlinear conditions.