The mass-mixed case for normalized solutions to NLS equations in dimension two

We are concerned with positive normalized solutions (u, lambda) is an element of H1 (& Ropf;2) & times; & Ropf; to the following semi-linear Schr & ouml;dinger equations-Delta u+lambda u=f(u),inR2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\Delta u+\lambda u=f(u),\quad\text{in}\,\mathbb{R}<^>{2},$$\end{document}satisfying the mass constraint integral R2|u|2dx=c2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int_{\mathbb{R}<^>{2}}\vert u\vert<^>{2}{\rm{d}}x=c<^>{2}$$\end{document}. We are interested in the so-called mass-mixed case in which f has L2-subcritical growth at zero and critical growth at infinity, which in dimension two turns out to be of exponential rate. Under mild conditions, we establish the existence of two positive normalized solutions provided the prescribed mass is sufficiently small: one is a local minimizer and the second one is of mountain-pass type. We also investigate the asymptotic behavior of solutions approaching the zero-mass case, namely when c -> 0+.