The Existence And Multiplicity Of The Solutions To A Class Of Fractional P-Laplace Equation

Fractional p-Laplace operators are a class of nonlocal elliptic operators,these operators are often applied to different practical problems,such as optimization prob-lems,phase transformation,semipermeable membrane problems.Variational method is commonly used to solve this kind of operator.But in this paper,the existence and mul-tiplicity of the solution of a class of fractional order p-Laplace equations is studied by Morse theory.The following are the contents:Firstly,we introduce the related knowledge of fractional order p-Laplace operator,(?)Then some critical conditions are given.According to the corresponding conditions,the implicit function theorem and the embedding theorem are used to discuss the critical groups at the zero point and at the infinity.Finally,the existence and multiplicity of the solution of the fractional order p—Laplace equation are proved by the application of the Morse theory with different critical groups.