Normalized solutions and mass concentration for supercritical nonlinear Schrodinger equations

In this paper,we deal with the existence and concentration of normalized solutions to the supercritical nonlinear Schrodinger equation {-Δu+V(x)u=μ_(q)u+a|u|^(q)u in R^(2),∫_(R^(2))|u|^(2)dx=1,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 v_(q).Furthermore,we study the limiting behavior of u_(q) and v_(q) when q→2_(+).Particularly,we describe precisely the blow-up formation of the excited state v_(q).