The Existence And Uniqueness Of Solution For A Singular Elliptic Differential Equation

SSingular elliptic differential equation is an important part of partial differential equation theory and an important tool to describe natural phenomena and explain natural laws.It has been widely studied in theory and application field.If we extend the nonsingular elliptic equation to singular elliptic equation and add singular terms,the problem will lose its compactility and the equation will lose its translation invariance.The mimimizers of the energy functional to the equation may not be the solution of the original equation.These characteristics make our general study of singular elliptic differential equations very meaningful.In this paper,we are concerned about the solution for the following singular elliptic differential equation(?),where N≥3,0 < q < 1,λ > 0,p > 1,μ > 0 is a parameter and h is a function satisfying certain conditions.When λ >N/2,the equation has a unique positive solution.When λ≤N/2,the equation has no positive solution.Based on variational argument and perturbation method,we obtain the existence,nonexistence and uniqueness of the positive solution for the singular elliptic differential equation.Then,we proof the above results by the method of numerical approximation.The specific contents of this paper are as follows:Firstly,we give a survey to the development of singular elliptic differential equations and the main work of this paper.Secondly,we make variational treatment for the singular ellliptic differential equation and define the sapces and norms which will be used.For the following proof,we describe the expressions of the principle eigenvalue of the linear equation for the singular ellliptic differential equation and the energy functional.Next,by using variational argument and perturbation method,we prove in turn that the energy functional of singular elliptic equation is lower bounded and forced,and the minimizer is still the solution of the elliptic differential equation under the singular case.On the basis of the above series of proofs,we obtain the existence for the singular elliptic differential equation as λ >N/2.Then,we prove the uniqueness of the positive solution by contradiction.Finally,from a experiment point of view,we obtain the numerical solution for the above singular elliptic differential equation by means of neural network under some special cases,which enrich the theoretical results.Our conclusions partially extend the results corresponding to the nonsingular case.