Remarks on Normalized Solutions for L2-Critical Kirchhoff Problems

We study a constraint minimization problem on S-c = {u epsilon H-1 c, c epsilon (0, c*)} for the following L-2-critical Kirchhoff type functional: E-alpha(u) =a/2 integral(RN) vertical bar del u vertical bar(2) dx + b/4 (integral(RN) vertical bar del vertical bar(2) dx)(2) + 1/alpha+2 integral(RN) V (x) vertical bar u vertical bar(alpha+2) dx - N/2N+8 integral(R2) vertical bar u vertical bar(2N+8/N) dx where N <= 3, a, b > 0 are constants, a epsilon [0, 8/N) and V(x) epsilon L infinity (R-N) is a suitable potential. We prove that the problem has at least one minimizer if alpha epsilon [2, 8/N) and the energy of the minimization problem is negative. Moreover, some non-existence results are obtained when the energy of the problem equals to zero.