Existence of normalized solutions to nonlinear Schrödinger equations with potential on lattice graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911041440448512 |
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| author | Guan, Weiqi |
| author_facet | Guan, Weiqi |
| contents | We study the existence of ground state normalized solution of the following Schrödinger equation: \begin{equation*} \begin{cases} -Δu+V(x)u+λu=f(x,u), & x\in\mathbb{Z}^d \\ \Vert u\Vert_2^2=a \end{cases} \end{equation*}
where $V(x)$ is trapping potential or well potential, $f(x,u)$ satisfies Berestycki-Lions type condition and other suitable conditions. We show that there always exists a threshold $α\in[0,\infty)$ such that there do not exist ground state normalized solutions for $a\in (0,α)$, and there exists a ground state normalized solution for $a\in(α,\infty)$. Furthermore, we prove sufficient conditions for the positivity of $α$ that $α=0$ if $f(x,u)$ is mass-subcritical near 0, and $α>0$ if $f(x,u)$ is mass-critical or mass-supercritical near 0. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04204 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence of normalized solutions to nonlinear Schrödinger equations with potential on lattice graphs Guan, Weiqi Analysis of PDEs We study the existence of ground state normalized solution of the following Schrödinger equation: \begin{equation*} \begin{cases} -Δu+V(x)u+λu=f(x,u), & x\in\mathbb{Z}^d \\ \Vert u\Vert_2^2=a \end{cases} \end{equation*} where $V(x)$ is trapping potential or well potential, $f(x,u)$ satisfies Berestycki-Lions type condition and other suitable conditions. We show that there always exists a threshold $α\in[0,\infty)$ such that there do not exist ground state normalized solutions for $a\in (0,α)$, and there exists a ground state normalized solution for $a\in(α,\infty)$. Furthermore, we prove sufficient conditions for the positivity of $α$ that $α=0$ if $f(x,u)$ is mass-subcritical near 0, and $α>0$ if $f(x,u)$ is mass-critical or mass-supercritical near 0. |
| title | Existence of normalized solutions to nonlinear Schrödinger equations with potential on lattice graphs |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.04204 |