Concentration behavior of normalized ground states for mass critical Kirchhoff equations in bounded domains

TL;DR

This paper investigates the limiting behavior and concentration phenomena of normalized ground states for a mass critical Kirchhoff equation in bounded domains, revealing conditions for existence, profile limits, and uniqueness as a parameter approaches zero.

Contribution

It provides a detailed analysis of the existence, asymptotic profiles, and local uniqueness of minimizers for the Kirchhoff equation with a critical nonlinearity in bounded domains.

Findings

Existence of minimizers if and only if a>0.

Mass concentration occurs at inner points or near boundary as a→0.

Local uniqueness of minimizers at a single concentration point.

Abstract

In present paper, we study the limit behavior of normalized ground states for the following mass critical Kirchhoff equation $$ \left\{\begin{array}{ll} -(a+b\int_{\Omega}|\nabla u|^2\mathrm{d}x)\Delta u+V(x)u=\mu u+\beta^*|u|^{\frac{8}{3}}u &\mbox{in}\ {\Omega}, \\[0.1cm] u=0&\mbox{on}\ {\partial\Omega}, \\[0.1cm] \int_{\Omega}|u|^2\mathrm{d}x=1, \\[0.1cm] \end{array} \right. $$ where $a\geq0$, $b>0$, 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 $-2\Delta u+\frac{1}{3}u-|u|^{\frac{8}{3}}u=0.$ 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 $\Omega$, 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 $\Omega$. In particular, we further establish the local uniqueness of the minimizer if it is concentrated at a unique inner point.