Solvability Of Elliptic Equations(Systems)

This paper is divided into four parts.The first part is the introduction,which introduces the main method of solving the boundary value problem of elliptic equation（group）and the main degree courses learned during the graduate school.The second part is the first chapter,which studies the solvability of boundary value problems of biharmonic equations.（?）Here Ω?Rn（n≥ 1）is abounded smooth convex region,f（x,s）,g（x,s）is a function defined on Ω× R+.Suppose the function in problem（1.1）satisfies（or partially satisfies）the following conditions:（A₁）f（x,s）,g（x,s）is a positive continuous function on Ω×R+,and（?）f（x,s）>0,（?）（x,s）>0,x∈Ω.（A₂）f（x,s）/s and g（x,s）/s decrease with respect to x∈Ω,s∈（0,+∞）.（A₃）For x∈Ω,（x,s）and g（x,s）are strictly monotonically increasing with respect to s∈（0,+∞）.（A₄）M<λ1,where λ1 is the first eigenvalue of the zero boundary value problem of the operator-Δ in Ω,where M=max{1,‖a（x）‖,‖b（x）‖｝,‖a（x）‖=（?）a（x）,‖b（x）‖（?）（x）here a（x）=（?）f（x,s）/s,b（c）=（?）g（x,s）/s.Proves the following theorems by using the certainty bound principle,Green’s first identity,Poincare inequality and fixed point theorem:Theorem 1.3（existence of solution）If condition（A₁）,（A₂）,（A₄）is true,so problem（1.1）has at least one bounded positive solution.Theorem 1.4（uniqueness of solution）If Condition （A₁）,（A₂）,（A₃）is true,so problem（1.1）has at most one bounded positive solution.The third part is the second chapter,the boundary value problem of semilinear elliptic equations is studied.（?）Here Ω?Rn is a bounded smooth region,and the existence of positive solutions for boundary value problems of biharmonic equations with positive small parameters （?） Where λ is the positive parameter,and the proof of the existence theorem of the positive solution is given.Let-Δu=v,then（2.4）becomes:（?）Suppose the function in problem（2.3）satisfies the following conditions:（B₁）f（x,s）,g（x,s）is a non-negative H?lder continuous function on Ω×[0,+∞).（B₂）f（x,s）,g（x,s）increases singly over[0,+∞)with respect to s,g（x,0）>0.The following theorems are proved by using the method of upper and lower solution and the fixed point theorem:Theorem 2.2（Existence of solutions to the first side value problem of elliptic equations）if（B₁）,（B₂）is true and problem（2.3）has a pair of upper and lower solutions,then problem（2.3）has at least one set of solutions（u,v）×[C²（Ω）]².Theorem 2.3（Existence theorem for the solution of the first side value problem of the biharmonic equation）When λ is a sufficiently small positive number and condition（B₁）,（B₂）is true,there is a bounded positive solution to problem（2.12）,and thus a positive solution to the biharmonic boundary value problem（2.4）.The fourth part is the summary and future research assumption.This article is briefly summarized,including the innovation of the article and the direction of future efforts.