Study On The Solution Of Two Kinds Of Kirchhoff Type Fractional Order Equations With Singularity

As an important research direction in the field of partial differential equations,the research on solutions to Kirchhoff problems based on classical Laplace operators and fractional Laplace operators has attracted more and more scholars’ attention in the past decade.Due to the extensive applications of fractional partial differential equations in various fields of mathematics and physics,the research on this type of equation,especially the solution of Kirchhoff type fractional partial differential equations,some progress has been made.It is worth noting that the existence of singular terms makes the study of fractional order partial differential equations more complex,and the research results need to be enriched.Therefore,this article will systematically study the existence and uniqueness of solutions for two types of Kirchhoff type fractional Laplace equations with singular terms.The specific research content is as follows:Firstly,this article briefly introduces the definitions and properties of fractional Sobolev spaces and fractional Sobolev type spaces,and introduces Ekeland variational principles,Nehari manifolds and describes some important conclusions on Lebesgue spaces.Then,this article investigates the existence and uniqueness of solutions for Kirchhoff type fractional Laplace equations with strong singular terms.The existence of strong singular terms makes the corresponding energy functional Fréchet derivative of the equation non-existent,which makes it impossible to directly discuss the extreme points of the energy functional using the Nehari manifold method.In order to overcome this difficulty,this paper constructs two constraint sets with the idea of Nehari manifold,studies the properties of energy functional on the constraint sets,and gives the necessary and sufficient conditions for the existence of unique positive solutions of the equation by applying the existence theorem of implicit function and Ekeland variational principle.Finally,this article discusses the existence and uniqueness of solutions for Kirchhoff type fractional Laplace equations with weak singular terms.By using the minimax method,it is proven that the equation has a unique positive solution.By studying the strong singular term and weak singular term equations,we can more intuitively understand the difference between strong and weak indicators in studying the existence and uniqueness of solutions.