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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2501.04893 |
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| _version_ | 1866913642570579968 |
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| author | Lin, Xiaolu Lv, Zongyan |
| author_facet | Lin, Xiaolu Lv, Zongyan |
| contents | In this paper, we consider the existence and multiplicity of prescribed mass solutions to the following nonlinear Schrödinger equation with general nonlinearity: Mass super-critical case: \[\begin{cases} -Δu+V(x)u+λu=g(u),\\ \|u\|_2^2=\int|u|^2\mathrm{d}x=c, \end{cases} \] both on large bounded smooth star-shaped domain $Ω\subset\mathbb{R}^N$ and on $\mathbb{R}^N$, where $V(x)$ is the potential and the nonlinearity $g(\cdot)$ considered here are very general and of mass super-critical. The standard approach based on the Pohozaev identity to obtain normalized solutions is invalid as the presence of potential $V(x)$. In addition, our study can be considered as a complement of Bartsch-Qi-Zou (Math Ann 390, 4813--4859, 2024), which has addressed an open problem raised in Bartsch et al. (Commun Partial Differ Equ 46(9):1729--1756, 2021). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04893 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Normalized Solutions on large smooth domains to the Schrödinger equation with potential and general nonlinearity: Mass super-critical case Lin, Xiaolu Lv, Zongyan Analysis of PDEs In this paper, we consider the existence and multiplicity of prescribed mass solutions to the following nonlinear Schrödinger equation with general nonlinearity: Mass super-critical case: \[\begin{cases} -Δu+V(x)u+λu=g(u),\\ \|u\|_2^2=\int|u|^2\mathrm{d}x=c, \end{cases} \] both on large bounded smooth star-shaped domain $Ω\subset\mathbb{R}^N$ and on $\mathbb{R}^N$, where $V(x)$ is the potential and the nonlinearity $g(\cdot)$ considered here are very general and of mass super-critical. The standard approach based on the Pohozaev identity to obtain normalized solutions is invalid as the presence of potential $V(x)$. In addition, our study can be considered as a complement of Bartsch-Qi-Zou (Math Ann 390, 4813--4859, 2024), which has addressed an open problem raised in Bartsch et al. (Commun Partial Differ Equ 46(9):1729--1756, 2021). |
| title | Normalized Solutions on large smooth domains to the Schrödinger equation with potential and general nonlinearity: Mass super-critical case |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.04893 |