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 $v_q$.
Categories: math.AP