Concentration behavior of normalized ground states for mass critical Kirchhoff equations in bounded domains
Year: 2025
Type: article
Abstract: In present paper, we study the limit behavior of normalized ground states for the following mass critical Kirchhoff equation \begin{align*}\left\{\begin{array}{ll}-(a+b\int_{\Omega}|\nabla u|^2\mathrm{d}x)\Delta u+V(x)u=\mu u+\beta^*|u|^{\frac{8}{3}}u &\text{in}\ {\Omega}, \\[0.1cm]u=0&\text{on}\ {\partial\Omega}, \\[0.1cm]\int_{\Omega}|u|^2\mathrm{d}x=1, \\[0.1cm]\end{array}\right.\end{align*} where $a\geq 0$ , b > 0, the function V ( x ) is a trapping potential in a bounded domain $\Omega\subset\mathbb R^3$ , $\beta^*:=\frac{b}{2}|Q|_2^{\frac{8}{3}}$ and Q is the unique positive radially symmetric solution of equation $-2\Delta u+\frac{1}{3}u-|u|^{\frac{8}{3}}u=0$ in $\mathbb R^3.$ We consider the existence of constraint minimizers for the associated energy functional involving the parameter a . The minimizer corresponds to the normalized ground state of above problem, and it exists if and only if a > 0. Moreover, when V ( x ) attains its flattest global minimum at an inner point or only at the boundary of Ω, we analyze the fine limit profiles of the minimizers as $a\searrow 0$ , including mass concentration at an inner point or near the boundary of Ω. In particular, we further establish the local uniqueness of the minimizer if it is concentrated at a unique inner point.
Authors: Shubin Yu, Chen Yang, Chun-Lei Tang
Institution: Southwest University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G3840989709
OpenAlex ID: W4415778102