Normalized solutions for nonlinear Schr\"odinger equations with $L^2$-critical nonlinearity

TL;DR

This paper investigates the existence and non-existence of normalized solutions to a nonlinear Schrödinger equation with $L^2$-critical nonlinearity, focusing on the effects of sublinear perturbations at infinity.

Contribution

It establishes the existence of solutions at critical mass for sublinear perturbations and proves non-existence under certain monotonicity conditions.

Findings

Existence of positive solutions at critical mass when perturbation is sublinear.

Non-existence results for perturbations not sublinear under specific conditions.

Characterization of solution existence based on growth conditions of the nonlinearity.

Abstract

We study the following nonlinear Schr\"odinger equation and we look for normalized solutions $(\mu,u)\in {\bf R}\times H^1({\bf R}^N)$ for a given $m>0$ and $N\geq 2$ \[ -\Delta u + \mu u = g(u)\quad \text{in}\ {\bf R}^N, \qquad \frac{1}{2}\int_{{\bf R}^N} u^2 dx = m. \] We assume that $g$ has an $L^2$-critical growth, both at the origin and at infinity. That is, for $p=1+\frac{4}{N}$, $g(s)=|s|^{p-1}s +h(s)$, $h(s)=o(|s|^p)$ as $s\sim 0$ and $s\sim\infty$. The $L^2$-critical exponent $p$ is very special for this problem; in the power case $g(s) = |s|^{p-1}s$ a solution exists only for the specific mass $m=m_1$, where $m_1=\frac{1}{2}\int_{{\bf R}^N}\omega_1^2\, dx$ is the mass of a least energy solution $\omega_1$ of $-\Delta \omega+\omega=\omega^p$ in ${\bf R}^N$. We prove the existence of a positive solution for $m=m_1$ when $h$ has a sublinear growth at infinity, i.e., $h(s)=o(s)$ as $s\sim\infty$. In contrast, we show non-existence results for $h(s)\not=o(s)$ ($s\sim 0$) under a suitable monotonicity condition.