Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.10258 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909254443597824 |
|---|---|
| author | Cassani, Daniele Huang, Ling Tarsi, Cristina Zhong, Xuexiu |
| author_facet | Cassani, Daniele Huang, Ling Tarsi, Cristina Zhong, Xuexiu |
| contents | \noindent We are concerned with positive normalized solutions $(u,λ)\in H^1(\mathbb{R}^2)\times\mathbb{R}$ to the following semi-linear Schrödinger equations $$ -Δu+λu=f(u), \quad\text{in}~\mathbb{R}^2, $$ 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 $c\to 0^+$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10258 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The mass-mixed case for normalized solutions to NLS equations in dimension two Cassani, Daniele Huang, Ling Tarsi, Cristina Zhong, Xuexiu Analysis of PDEs \noindent We are concerned with positive normalized solutions $(u,λ)\in H^1(\mathbb{R}^2)\times\mathbb{R}$ to the following semi-linear Schrödinger equations $$ -Δu+λu=f(u), \quad\text{in}~\mathbb{R}^2, $$ 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 $c\to 0^+$. |
| title | The mass-mixed case for normalized solutions to NLS equations in dimension two |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.10258 |