| This paper is divided into four chapters.The existence and uniqueness of solutions of nonlinear partial differential equations(systems)are mainly researched.The first chapter is the introduction.The research background and significance of partial differential equations are introduced.The research status of partial differential equations at home and abroad are briefly described.Also,the main contents and the methods used in this paper are explained.In second chapter,we investigate the polyharmonic equations boundary problem of solvability.Here Ω(?)Rn is a bounded smooth area.The study of polyharmonic equations boundary problems is one of the important contents in the study of elliptic systems boundary problems.For example,literature[1]researched the problem generalized solutions.In Sobolev space,the existence of generalized solutions under certain conditions have been proved by using embedding theorem and other theories.Literature[2]discussed the problem D (?) R2 is a bounded Lipschitz area.The above problem of unique integral means solution has been deduced by using multilayer S potential in this paper.Literature[3-5]have researched the existence of solutions of polyharmonic equations boundary problems.The issue discussed in this chapter is the polyharmonic equations boundary problem.In this equation,the nonlinear term f(x,u)conditions are different from the problem researched in literature[1]and the problem discussed in literature[2].The conditions required to be fulfilled are also relatively common.Therefore,the scope of discussion of this kind of problems are relatively wide.In this chapter,a class of polyharmonic equations boundary problems are transformed into elliptic systems boundary problems by using variable substitution.Then the existence of positive solutions of elliptic systems boundary problems and this class of polyharmonic equations boundary problems have been proved by using the fixed point theorem,extreme value principle,Green identity and other theoretical methods.Also,the uniqueness of solution is discussed.As an application of the main theorem,two specific examples are given.In third chapter,we investigate the semilinear elliptic system ai(x)(i=1,2,3,4)is a non-negative Holder continuous function on Ω.Ω is a bounded domain with a hole in Rn.The inner and outer boundaries in Ω are recorded as Γ1 and Γ2.Also,the boundary of Ω is smooth.b>0 is a positive parameter.In early literature[6],the positive solutions of semilinear elliptic equations boundary problem for the following has researched This paper concluded that there is a positive number b*.When b<b*,the above problem has positive solutions.When b>b’,the above problem has no solution.In literature[7],the semilinear elliptic equations boundary problem on the ring area for the following has researched and the same conclusion are obtained.The nonlinear term f(u)is a convex function,also superlinear at 0 and ∞.In literature[8],the elliptic boundary problem on the ring area Ω={x∈RN |r1<|x|<r2}has researched existence of radial solutions.The nonlinear term f(r,u,η)obtained the existence of radial solutions to this problem in the case of u,η superlinear growth.Literature[9-10]have researched the existence of solutions of semilinear elliptic systems boundary problems.The issue discussed in this chapter is the semilinear elliptic systems boundary problem in a bounded domain with a hole.This issue is more complex than single equation boundary problems.Also,results of the existence and uniqueness of solutions of this kind of problems are obtained.In this chapter,the existence of a class of semilinear elliptic systems with first boundary condition are discussed in a bounded domain with a hole.Each equation in this system have contained linear part and nonlinear part about unknown functions.Systems boundary problems are transformed into vector equations boundary problems by using variable substitution.Then the existence of positive solutions has been proved by using the fixed point theorem,Green’s first identity,Poincare inequality and other theoretical methods.Also under certain conditions,the uniqueness of solution is discussed.As an application of the main theorem,two specific examples of the existence and uniqueness of positive solutions are given.The fourth chapter is the conclusion and prospect part.A brief summary in this paper is given.The work done in this paper is introduced.Also,the future research work of partial differential equations is prospected. |
