A Class Of Fractional Kirchhoff Equations And Its Singular Perturbation Problems

In this paper,we construct a variational functional for a class of fractional Kirchhoff equation and its related perturbation problems under certain assumptions.The study of existence and multiplicity of solutions for the fractional Kirchhoff equation and its related perturbation problems is transformed into the study of existence and multiplicity of critical points for the corresponding variational functional.According to the characteristics of nonlocal term and Kirchhoff operator,we transform the nonlocal problem into local problem by Dirichlet-Neumann mapping to study the uniqueness and non-degeneracy of solutions of fractional Kirchhoff equation.When the perturbed parameter ε>0 is small enough,we can obtain the existence of solutions for singularly perturbed fractional Kirchhoff problems by using non-degenerate results and Lyapunov-Schmidt reduction method.Under some suitable assumptions of potential Ⅴ,based on ljusternik-schnirelmann theory,we associate the number of positive solutions with the topological degree of the set where V reaches its minimum,and then obtain the multiplicity of positive solutions.The results obtained in this paper extend the integer order Kirchhoff equation to the fractional order,which extends the application of the variational method to Kirchhoff problem with fractional Laplacian to a certain extent.