Multiplicity Solutions For A Elliptic Equation

It is a core of the theory of research field of differential equation to study multiplicityresults for differential equations under boundary condition. Also, it is one of the key subjectsof the research contents of this field.It has backgrounds of deep physics and mechanics to utilize the theories of topologicaldegree, variational reduction method and critical point principle to investigate solvability andmultiplicity results of differential equations under boundary condition. Solving the problemneeds topological and geometrical properties of classical space. At the same time, the set-tlement of the problem drives a lot of new production and development of tool in nonlinearanalysis, and has shown a multi-disciplinary research field that blends each other. Througha large number of mathematicians' efforts, the theory of this field has already formed a kindof typical treatment method for partial differential equations.This text utilizes these methodsmainly, study solvability of the several differential equations and multiplicity of solutions onthe basis of forefathers.1. The first chapter introduces several symbols, definitions and some results about ellipticequation,which have been obtained by using the theories of topological degree, varia-tional reduction method,Mountain pass theorem and critical point principle.2. In the second and third chapter, we mainly investigate a fourth order semilinear ellipticequation under Dirichlet boundary value.â–³²u+câ–³u=b₁[(u+1)+-1]+b₂u^-在Ω中u=0,â–³u=0在(?)Ω上We mainly investigate the solutions of the equation in different six regions of (b₁, b₂) whenλ₁＜c＜λ₂. Firstly, we introduce the Sobolev space spanned by the eigenfunctions ofoperatorâ–³²+câ–³. Secondly we prove the associated functional G satisfies (PS)condition;Finally, we investigate the multiplicity solutions for the equation in six regions of (b₁, b₂);...