Existence of normalized solutions to nonlinear Schrödinger equations on lattice graphs
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911034903625728 |
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| author | He, Zhentao Ji, Chao Tao, Yifan |
| author_facet | He, Zhentao Ji, Chao Tao, Yifan |
| contents | In this paper, using a discrete Schwarz rearrangement on lattice graphs developed in \cite{DSR}, we study the existence of global minimizers for the following functional $I:H^1\left(\mathbb{Z}^N\right)\to \R$, $$I(u)=\frac{1}{2} \int_{\mathbb{Z}^N}|\nabla u|^2 \,dμ-\int_{\mathbb{Z}^N} F(u)\, dμ,$$ constrained on $S_m:=\left\{u \in H^1\left(\mathbb{Z}^N\right) \mid\|u\|_{\ell^2\left(\mathbb{Z}^N\right)}^2=m\right\}$, where $N \geq 2$, $m>0$ is prescribed, $f \in C(\mathbb{R}, \mathbb{R})$ satisfying some technical assumptions and $F(t):=\int_0^t f(τ) \,dτ$. We prove the following minimization problem $$ \inf_{u \in S_m} I(u) $$
has an excitation threshold $m^*\in [0,+\infty]$ such that \begin{equation*}
\inf_{u \in S_m} I(u)<0 \quad \text{if and only if } m>m^*. \end{equation*} Based primarily on $m^* \in (0,+\infty)$ or $m^*=0$, we classify the problem into three different cases: $L^2$-subcritical, $L^2$-critical and $L^2$-supercritical. Moreover, for all three cases, under assumptions that we believe to be nearly optimal, we show that $m^*$ also separates the existence and nonexistence of global minimizers for $I(u)$ constrained on $S_{m}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_01591 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence of normalized solutions to nonlinear Schrödinger equations on lattice graphs He, Zhentao Ji, Chao Tao, Yifan Analysis of PDEs 35A15, 35Q55, 35R02 In this paper, using a discrete Schwarz rearrangement on lattice graphs developed in \cite{DSR}, we study the existence of global minimizers for the following functional $I:H^1\left(\mathbb{Z}^N\right)\to \R$, $$I(u)=\frac{1}{2} \int_{\mathbb{Z}^N}|\nabla u|^2 \,dμ-\int_{\mathbb{Z}^N} F(u)\, dμ,$$ constrained on $S_m:=\left\{u \in H^1\left(\mathbb{Z}^N\right) \mid\|u\|_{\ell^2\left(\mathbb{Z}^N\right)}^2=m\right\}$, where $N \geq 2$, $m>0$ is prescribed, $f \in C(\mathbb{R}, \mathbb{R})$ satisfying some technical assumptions and $F(t):=\int_0^t f(τ) \,dτ$. We prove the following minimization problem $$ \inf_{u \in S_m} I(u) $$ has an excitation threshold $m^*\in [0,+\infty]$ such that \begin{equation*} \inf_{u \in S_m} I(u)<0 \quad \text{if and only if } m>m^*. \end{equation*} Based primarily on $m^* \in (0,+\infty)$ or $m^*=0$, we classify the problem into three different cases: $L^2$-subcritical, $L^2$-critical and $L^2$-supercritical. Moreover, for all three cases, under assumptions that we believe to be nearly optimal, we show that $m^*$ also separates the existence and nonexistence of global minimizers for $I(u)$ constrained on $S_{m}$. |
| title | Existence of normalized solutions to nonlinear Schrödinger equations on lattice graphs |
| topic | Analysis of PDEs 35A15, 35Q55, 35R02 |
| url | https://arxiv.org/abs/2507.01591 |