Limiting behavior and local uniqueness of normalized solutions for mass critical Kirchhoff equations

In this paper, we consider the L-2-norm prescribed ground states of the Kirchhoff equations involving mass critical exponent with a Lagrange multiplier mu is an element of R, {-(a+b integral(RN) vertical bar del u vertical bar(2) dx) Delta u + V(x)u = mu u + u (R/N + 1) in R-N, integral(RN) vertical bar u vertical bar(2) dx = c(2), c > 0, where a >= 0, b > 0, N = 1, 2, 3, and the function V (x) is an element of L-loc(infinity)(R-N) is a trapping potential satisfying min(x is an element of RN) V (x) = 0, V (x) ->+infinity as vertical bar x vertical bar -> +infinity. It has been shown by researchers that there exists a couple of ground state solution (u(a), mu(a)) to (0.1) if c = c(*) := (b parallel to Q parallel to(8/N)(2)/2)(N/8-2N) for small a > 0, where Q > 0 is the unique radially symmetric positive solution of equation 2 Delta Q + N-4/N Q+Q(8/N+1) = 0 in R-N. We devote to the refined limiting profiles of u(a) as a -> 0 by using energy estimates and blow-up analysis. In order to get the concentration behavior of u(a), we first study the existence and non-existence of solutions to a degenerate Kirchhoff equation i.e. the case a = 0 in (0.1). At last, we investigate the local uniqueness of ground states u(a) included by concentration.