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

TL;DR

This paper proves the existence of two positive normalized solutions for a 2D nonlinear Schrödinger equation with mixed growth conditions, analyzing their behavior as the mass approaches zero.

Contribution

It establishes the existence of two solutions under small mass constraints in the mass mixed case with exponential critical growth in two dimensions.

Findings

Existence of two positive normalized solutions for small mass.

One solution is a local minimizer, the other is mountain pass type.

Asymptotic behavior as mass approaches zero is characterized.

Abstract

\noindent We are concerned with positive normalized solutions $(u,\lambda)\in H^1(\mathbb{R}^2)\times\mathbb{R}$ to the following semi-linear Schr\"{o}dinger equations $$ -\Delta u+\lambda u=f(u), \quad\text{in}~\mathbb{R}^2, $$ satisfying the mass constraint $$\int_{\mathbb{R}^2}|u|^2\, dx=c^2\ .$$ We are interested in the so-called mass mixed case in which $f$ has $L^2$-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 $c\to 0^+$.