Mass concentration of minimizers for L2-subcritical Kirchhoff energy functional in bounded domains

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