Existence And Uniqueness Of Solutions For A Class Of Fractional Kirchhoff Equations

Kirchhoff equation is derived from the free vibration of elastic strings.Since Lions obtained the form of equation by using the framework of functional analysis,it has attracted extensive attention from mathematical researchers.But,variational methods seem difficult to use because of the existence of non-local term in Kirchhoff equation.Moreover,the fractional Laplacian operator is also non-local.Hence,it seems that the fractional Kirchhoff equation is more challenging.This paper mainly studies the existence and uniqueness of ground state solutions of a class of fractional Kirchhoff equations by using the energy estimates technique and the constrained variational theory.The specific research work of this paper is as follows:Firstly,the power law nonlinear fractional Kirchhoff equation was considered.Using energy estimates and the constrained variational theory,the existence of the ground state solution of nonlinear fractional Kirchhoff equation was obtained.In this part,we gave a complete classification with respect to the exponential of the nonlinear term for existence of ground state.Moreover,the uniqueness of the ground state solution is obtained by using the energy estimates and the scaling technique.Secondly,the existence of Schwarz symmetric minimizers for a class of fractional Kirchhoff type constrained variational problems with general nonlinear terms was studied.Since the nonlinear term is a general case,the scaling technique used for the exponential nonlinear term is no longer applicable,and the compactness in the dichotomy part of the concentration-compactness principle is difficult to exclude.In this section,the existence and nonexistence of Schwarz symmetric minimizers for fractional Kirchhoff constrained variational problem with general nonlinear terms were described by using convexity analysis of functional and detailed energy estimate.In a word,the existing relevant results are mainly about the solutions of the fractional Laplace equation with general nonlinear terms or the existence of ground state solutions of the Kirchhoff equation.In this paper,the existence and uniqueness of ground state solutions of a class of fractional Kirchhoff equations are mainly considered.The equation becomes more complex,the method needs to be improved since the existence of non-local term.