Normalized solutions for nonlinear Schrödinger equations with $L^2$-critical nonlinearity

Abstract

We study the following nonlinear Schrödinger equation and we look for normalized solutions $(\hat{I}¼,u)\in \mathbf{R}\times H^1({\mathbf{R}}^N)$ for a given $m>0$ and $N\ge 2$ \[ -Δu + μu = g(u)\quad \text{in}\ {\bf R}^N, \qquad \frac{1}{2}\int_{{\bf R}^N} u^2 dx = m. \] 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 $m_1=\frac{1}{2}{\int }_{{\mathbf{R}}^N}\ddot{I}{‰}_1^{2}dx$ is the mass of a least energy solution $\ddot{I}{‰}_1$ of $-\hat{I}”\ddot{I}‰+\ddot{I}‰=\ddot{I}{‰}^p$ in ${\mathbf{R}}^N$. 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 $h(s)\ne o(s)$ ($s\sim 0$) under a suitable monotonicity condition.

60 pages; Proposition 1.2 and Corollary 1.3 updated; Section 2.4 added; typos corrected, a reference added