Normalized solutions and mass concentration for supercritical nonlinear Schr\"{o}dinger equations

In this paper, we deal with the existence and concentration of normalized solutions to the supercritical nonlinear Schr"{o}dinger equation \begin{equation*} \left{ \begin{array}{l} -\Delta u + V(x) u = \mu_q u + a|u|^q u \quad {\rm in}\quad $\mathbb{R}$^2,\ \int_{$\mathbb{R}$^2}|u|^2,dx =1,\ \end{array} \right. \end{equation*} where ${\mu }_q$ is the Lagrange multiplier. We show that for $q>2$ close to $2$, the equation admits two solutions: one is the local minimal solution $u_q$ and another one is the mountain pass solution $v_q$. Furthermore, we study the limiting behavior of $u_q$ and $v_q$ when $q\to 2_{+}$. Particularly, we describe precisely the blow-up formation of the excited state $v_q$.