Normalized solutions and mass concentration for supercritical nonlinear Schrödinger equations

Abstract

In this paper, we deal with the existence and concentration of normalized solutions to the supercritical nonlinear Schrödinger equation

$$\left\{ {\matrix{ { - {\rm{\Delta }}u + V\left( x \right)u = {\mu _q}u + a{{\left| u \right|}^q}u\,\,\,{\rm{in}}\,{\mathbb{R}^2},} \hfill \cr {\int_{{\mathbb{R}^2}} {{{\left| u \right|}^2}dx = 1,} } \hfill \cr } } \right.$$

where μ_q is the Lagrange multiplier. We show that for q > 2 close to 2, the problem admits two solutions: one is the local minimal solution u_q and the other one is the mountain pass solution υ_q. Furthermore, we study the limiting behavior of u_q and υ_q when q → 2₊. Particularly, we describe precisely the blow-up formation of the excited state υ_q.