Study On The Solution Of Fractional Kirchhoff Equation With Hardy Term

Since the classical Kirchhoff model was proposed,the Kirchhoff model based on the integer order Laplace operator has been widely used in non Newtonian mechanics,elasticity,materials science and other fields.However,with the deepening of research,physicists found that the integer order Laplacian operator can no longer accurately describe the change of chord tension at the fractional length,and thus proposed the fractional order Kirchhoff model.In the past decade,the study of solutions to fractional order Kirchhoff equations has attracted a lot of attention from mathematical workers.It is worth noting that the homogeneity of Hardy terms has a certain impact on the analytical properties of solutions to fractional order Kirchhoff equations.Therefore,this article will conduct in-depth research on the existence,uniqueness,and multiplicity of solutions to fractional order Kirchhoff equations with positive and negative Hardy terms.The specific research content is as follows:Firstly,this article discusses the research background and current development status of classical Kirchhoff problems and fractional order Kirchhoff problems,and provides the definitions of fractional order Sobolev spaces and fractional order Sobolev type spaces,as well as the Ekeland variational principle and other important lemmas required for this article’s discussionSecondly,this article discusses the existence and uniqueness of solutions for a class of strongly singular fractional Kirchhoff type equations with positive Hardy terms.The difficulty in studying this equation lies in the Hardy term and strong singular term.Therefore,this article uses fractional order Hardy inequality to make the Hardy term bounded,and uses the Nehari manifold idea to define a restricted set,so that strong singular terms can restore integrability on the restricted set.Furthermore,the Ekeland variational principle is applied to prove that the equation has a unique positive solutionFinally,this article investigates the multiplicity of solutions for weakly singular fractional order Kirchhoff type equations with negative Hardy terms.This equation is a variation of the previous equation,and the Kirchhoff term is more general and contains both negative Hardy and weakly singular terms in non-homogeneous terms.This article applies Ekeland’s variational principle,techniques for handling Nehari manifolds,and fractional Hardy inequality to prove that the equation has at least two positive solutions...