Normalized solutions for nonlinear Schrödinger equations with $L^2$-critical nonlinearity
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909535002689536 |
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| author | Cingolani, Silvia Gallo, Marco Ikoma, Norihisa Tanaka, Kazunaga |
| author_facet | Cingolani, Silvia Gallo, Marco Ikoma, Norihisa Tanaka, Kazunaga |
| contents | We study the following nonlinear Schrödinger equation and we look for normalized solutions $(μ,u)\in {\bf R}\times H^1({\bf R}^N)$ for a given $m>0$ and $N\geq 2$
\[ -Δu + μu = g(u)\quad \text{in}\ {\bf R}^N, \qquad
\frac{1}{2}\int_{{\bf R}^N} u^2 dx = m.
\] 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 $m_1=\frac{1}{2}\int_{{\bf R}^N}ω_1^2\, dx$ is the mass of a least energy solution $ω_1$ of $-Δω+ω=ω^p$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_23733 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normalized solutions for nonlinear Schrödinger equations with $L^2$-critical nonlinearity Cingolani, Silvia Gallo, Marco Ikoma, Norihisa Tanaka, Kazunaga Analysis of PDEs 35B33, 35B38, 35J20, 35Q55, 49J35, 35A01, 35J91, 47F10, 47J30 We study the following nonlinear Schrödinger equation and we look for normalized solutions $(μ,u)\in {\bf R}\times H^1({\bf R}^N)$ for a given $m>0$ and $N\geq 2$ \[ -Δu + μu = g(u)\quad \text{in}\ {\bf R}^N, \qquad \frac{1}{2}\int_{{\bf R}^N} u^2 dx = m. \] 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 $m_1=\frac{1}{2}\int_{{\bf R}^N}ω_1^2\, dx$ is the mass of a least energy solution $ω_1$ of $-Δω+ω=ω^p$ 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. |
| title | Normalized solutions for nonlinear Schrödinger equations with $L^2$-critical nonlinearity |
| topic | Analysis of PDEs 35B33, 35B38, 35J20, 35Q55, 49J35, 35A01, 35J91, 47F10, 47J30 |
| url | https://arxiv.org/abs/2410.23733 |