Mass concentration of minimizers for $L^2$-subcritical Kirchhoff energy functional in bounded domains

Abstract

We are concerned with $L^2$-constraint minimizers for the Kirchhoff functional $$E_b(u)={\int }_{\hat{I}}\text{c◯}|\nabla u{|}^2\mathrm{d}x+\frac{b}{2}{\left({\int }_{\hat{I}}\text{c◯}|\nabla u{|}^2\mathrm{d}x\right)}^2+{\int }_{\hat{I}}\text{c◯}V(x)u^2\mathrm{d}x-\frac{\hat{I}}{²}2{\int }_{\hat{I}}\text{c◯}|u{|}^4\mathrm{d}x,$$ where $b>0$, ${\hat{I}}^2>0$ and $V(x)$ is a trapping potential in a bounded domain $\hat{I}\text{c◯}$ of ${\mathbb{R}}^2$. As is well known that minimizers exist for any $b>0$ and ${\hat{I}}^2>0$, while the minimizers do not exist for $b=0$ and $β\geqβ^*$, where $β^*=\int_{\mathbb R^2}|Q|^2\mathrm{d}x$ and $Q$ is the unique positive solution of $-\hat{I}”u+u-u^3=0$ in ${\mathbb{R}}^2$. In this paper, we show that for $β=β^*$, the energy converges to 0, but for $β>β^*$, the minimal energy will diverge to $-\infty$ as $b\searrow 0$. Further, we give the refined limit behaviors and energy estimates of minimizers as $b\searrow 0$ for $β=β^*$ or $β>β^*$. For both cases, we obtain that the mass of minimizers concentrates either at an inner point or near the boundary of $\hat{I}\text{c◯}$, depending on whether $V(x)$ attains its flattest global minimum at an inner point of $\hat{I}\text{c◯}$ or not. Meanwhile, we find an interesting phenomenon that the blow-up rate when the minimizers concentrate near the boundary of $\hat{I}\text{c◯}$ is faster than concentration at an interior point if $β=β^*$, but the blow-up rates remain consistent if $β>β^*$.