Constraint minimizers of mass critical fractional Kirchhoff equations: concentration and uniqueness

This paper focuses on the constraint minimization problem associated with the fractional Kirchhoff equation {(a+b integral N-& Ropf;|(-Delta)(s/2)u|(2)dx)(-Delta)(s)u+|x|(2)u=mu u+beta u(8s/N)+1 in & Ropf;(N), integral N-& Ropf;|u|(2)dx=1, where s is an element of(N/4,1),N=2,3, a >= 0,b>0 are constants, mu is an element of R is the corresponding Lagrange multiplier and (-Delta)(s) is the fractional Laplacian operator, 8s/N+1 is the corresponding mass critical exponent. The purpose of this paper is threefold: to establish the existence and non-existence of the L-2-constraint minimizers to the degenerate fractional Kirchhoff problem, that is a = 0, to prove some classical concentration behaviors of constraint minimizers and to reveal the local uniqueness of constraint minimizers of above problem under double nonlocal effect. In particular, we will give some energy estimates, decay estimates and uniform regularity to find that the maximal point of constraint minimizer concentrates on the bottom point of the homogeneous potential. Furthermore, we introduce several new techniques based on the combination of the localization method of (-Delta)(s) and by establishing the nonlocal Pohoz & abreve;ev identity, which allow us to get over some new challenges due to the nonlocal property of (-Delta)(s) and the fact that integral N-R|(-Delta)(s/2)u|(2)dx(-Delta)(s)u does not vanish as a SE arrow 0. We believe that these techniques will have some potential applications in various related problems.