Properties of minimizers for mass critical Kirchhoff energy functionals in bounded domains
Helin Guo, Chenhao Wei, Lingling Zhao · Applicable Analysis · 2025
In this paper, we consider the properties of minimizers for the following constraint minimization problem: e(β):=infu∈SEβ(u),where S={u∈H01(Ω):∫Ω|u|2dx=1}, and the energy functional Eβ(⋅) is defined by Eβ(u):=b4(∫Ω|∇u|2dx)2+12∫ΩV(x)|u|2dx−Nβ8+2N∫Ω|u|8N+2dx,u∈H01(Ω).We prove that there is a threshold β∗>0 such that minimizers exist for 0<β<β∗ and the minimizer does not exist for β≥β∗. Moreover, we analyze the concentration behavior of minimizers as β↗β∗, which shows that the mass of minimizers must concentrate either at an inner point of Ω or near the boundary of Ω, depending on whether the potential V(x) attains its global minimum at an inner point or not of Ω.