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 $$E_b(u)={\int }_{\Omega }|\nabla u{|}^2\mathrm{d}x+\frac{b}{2}{\left({\int }_{\Omega }|\nabla u{|}^2\mathrm{d}x\right)}^2+{\int }_{\Omega }V(x)u^2\mathrm{d}x-\frac{\beta }{2}{\int }_{\Omega }|u{|}^4\mathrm{d}x,$$ where $b>0$, $\beta >0$ and $V(x)$ is a trapping potential in a bounded domain $\Omega$ of ${\mathbb{R}}^2$. As is well known that minimizers exist for any $b>0$ and $\beta >0$, while the minimizers do not exist for $b=0$ and $\beta \ge {\beta }^{\ast }$, where ${\beta }^{\ast }={\int }_{{\mathbb{R}}^2}|Q{|}^2\mathrm{d}x$ and $Q$ is the unique positive solution of $-\Delta u+u-u^3=0$ in ${\mathbb{R}}^2$. In this paper, we show that for $\beta ={\beta }^{\ast }$, the energy converges to 0, but for $\beta >{\beta }^{\ast }$, 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 $\beta ={\beta }^{\ast }$ or $\beta >{\beta }^{\ast }$. For both cases, we obtain that the mass of minimizers concentrates either at an inner point or near the boundary of $\Omega$, depending on whether $V(x)$ attains its flattest global minimum at an inner point of $\Omega$ or not. Meanwhile, we find an interesting phenomenon that the blow-up rate when the minimizers concentrate near the boundary of $\Omega$ is faster than concentration at an interior point if $\beta ={\beta }^{\ast }$, but the blow-up rates remain consistent if $\beta >{\beta }^{\ast }$.