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

In this paper, we consider the L-2-norm prescribed minimizer of the mass critical Kirchhoff type energy functional with a weight function a(x), E(u) = integral(RN) a(x)|del u|(2)dx + b/2(integral(RN) |del u|(2)dx)(2) - N/N+4 integral(RN) |u|(2N+8/N)dx, 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 NE arrow c(*) := (b||Q||(8/N)(2))(N/8-2N) is investigated, where Q > 0 is the unique radially symmetric positive solution of 2 Delta Q - (4/N - 1)Q + Q(8/N+1) = 0 in R-N. 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. (c) 2023 Elsevier Inc. All rights reserved.