Existence And Multiplicity Of Positive Solutions For A Class Of Elliptic Equations

In elliptic equations,the problem with the boundary value condition has al-ways attracted the attention of many scholars.In the field of mathematics,elliptic equations with nonlinear boundary constraints have a wide range of applications.Therefore,the study of the solutions of this kind of partial differential equations with nonlinear boundary values is also popular.In this thesis,we will introduce the study of the relevant properties of the solutions of elliptic equations with nonlinear boundary value conditions,and give the results of the existence and multiplicity of the solutions of the relevant research models.This thesis mainly includes three parts.The first part,we introduce the elliptic thermal explosion model with nonlin-ear boundary value in the bounded domain(?)in two-dimensional space R~2.In this model,we demonstrate the comparison principle under nonlinear boundary condi-tions based on limit conditions.Using the sub supersolution theorem,we prove the existence of the positive solution of K(x) and the monotonicity of the maximum so-lution on the parameter.More importantly,we have concluded that the solution is unique when K(x)<0.Because this is a nonlinear boundary value problem,we will have more limitations when using some general comparison principles.For this reason,we focus on proving the comparison principles applicable to nonlinear boundary value problems,making the proving process clearer.This is also an important breakthrough in this chapter.Secondly,we introduced nonlinear boundary value problems.and selected this high-dimensional space R~N(N≥3).At this time,we can choose to apply some appropriate embedding to prove whether the model solution exists.Then we proved that in high-dimensional bounded spaces,the multiplicity of solutions can be ob-tained under the nonlinear boundary value condition n·Δμ+g(μ)μ=0.Due to the change of dimension,some embedding theorems commonly used in two-dimensional space will lose some force,which makes our proof more complex.Therefore,in this chapter,we do not only use the upper and lower solutions to solve the multiplicity problem of solutions,but also use the variational method to construct the energy functional to make up for the shortcomings in the embedding process.Finally,we discuss the existence results of a class of nonlinear boundary value problems in critical cases.The usual A-R condition requires the index to be in a subcritical state.When the critical exponent is 2~*,for the elliptic equation with Dirichlet boundary,someone has studied the embedding of this part lacking com-pactness.For the nonlinear boundary value condition,and the index is critical,the lack of compactness is the difficulty that we want to focus on in chapter 5.There-fore,we use the variational method to construct the road connection to prove the existence result of the solution of the nonlinear boundary value problem when the critical exponent is 2~*=2N/N-2 and satisfies 1≤q<2~*-1.