Normalized bound state solutions to Kirchhoff equation with mass-critical exponent

This paper is concerned with the following L2-norm constrained Kirchhoff equation −1+b∫R3|∇u|2dxΔu+V(x)u+λu=|u|83u,x∈R3,u(x)∈H1(R3),u(x)≥0,∫R3|u|2dx=m2 in the mass-critical setting. The parameters b > 0 and m > 0 are given in advance, and the unknown parameter λ appears as a Lagrange multiplier. Here, the potential V(x) is positive, vanishing at infinity, and may have singular points. Under some explicit smallness assumptions on V(x), we show the existence of a bound state solution (v,λ)∈H1(R3)×R+ and the nonexistence of ground state solutions. To overcome the lack of compactness resulting from the L2-norm constraint, we introduce some new and subtle energy estimates that fully utilize the features of the mass critical exponent.