# 学术报告三十一：杨健夫—Normalized solutions and mass concentration for supercritical nonlinear Schrödinger equations

-\Delta u + V(x) u = \mu_q u + a|u|^q u \q in \mathbb{R}^2,

\int_{\mathbb{R}^2}|u|^2\,dx =1,\\

where $\mu_q$ is the Lagrange multiplier. We show that for q>2 close to 2,problem \eqref{eq:0.1} 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 $v_q$.