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) = ∫ R N a (x) | ∇ u | 2 d x + b 2 (∫ R N | ∇ u | 2 d x) 2 − N N + 4 ∫ R N | u | 2 N + 8 N d x , 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 ‖ 2 8 / N 2) N 8 − 2 N is investigated, where Q > 0 is the unique radially symmetric positive solution of 2 Δ 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. [ABSTRACT FROM AUTHOR]