Concentration and local uniqueness of minimizers for mass critical degenerate Kirchhoff energy functional

Abstract

In this paper, we consider the L2-norm prescribed minimizer of the mass critical Kirchhoff type energy functional with a weight function a(x),E(u)=∫RNa(x)|∇u|2dx+b2(∫RN|∇u|2dx)2−NN+4∫RN|u|2N+8Ndx,N=1,2,3. Making use of the Gagliardo–Nirenberg inequality, we firstly give the classification of existence and non-existence of minimizers. Then the mass concentration of minimizers as c↗c⁎:=(b‖Q‖28/N2)N8−2N is investigated, where Q>0 is the unique radially symmetric positive solution of 2ΔQ−(4N−1)Q+Q8N+1=0 in RN. It is surprise that the concentrating point of a minimizer is possibly determined by the weight function a(x). Finally, we analyze the local uniqueness of minimizers induced by concentration.