Constrained variational problems on perturbed lattice graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917275416657920 |
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| author | Guan, Weiqi |
| author_facet | Guan, Weiqi |
| contents | In this paper, we solve some constrained variational problems on perturbed lattice graphs $G$. The first problem addresses the existence of ground state normalized solutions to Schrödinger equations
\begin{equation*} \left\{
\begin{aligned}
&-Δ_{G} u+λu=\vert u\vert^{p-2}u,x\in G
&\Vert u\Vert_{l^2(G)}^2=a.
\end{aligned}
\right. \end{equation*} We prove that if the graph is obtained by deleting finite edges in lattice graphs while maintaining connectivity, then there exists a threshold $α_G\in[0,\infty)$ such that there do not exist ground state normalized solution if $0<a<α_G$, and there exists a ground state normalized solution if $a>α_G.$ If the graph is obtained by adding finite edges $E^{'}$ to lattice graphs, we prove that there exist $E^{'}$ and $a_1$ such that for all $a>a_1,$ there do not exist ground state normalized solutions.
The second problem concerns the existence of an extremal function for the Sobolev inequality. If the graph $G$ is obtained by deleting finite edges in lattice graphs while maintaining connectivity, for the Sobolev super-critical regime, we prove that there exists an extremal function. for the Sobolev critical regime, we prove that there exists $G$ such that extremal can be attained. If the graph is obtained by adding finite edges $E^{'}$ to lattice graphs, we prove that there exists $E^{'}$ such that there does not exist an extremal function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13978 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Constrained variational problems on perturbed lattice graphs Guan, Weiqi Analysis of PDEs In this paper, we solve some constrained variational problems on perturbed lattice graphs $G$. The first problem addresses the existence of ground state normalized solutions to Schrödinger equations \begin{equation*} \left\{ \begin{aligned} &-Δ_{G} u+λu=\vert u\vert^{p-2}u,x\in G &\Vert u\Vert_{l^2(G)}^2=a. \end{aligned} \right. \end{equation*} We prove that if the graph is obtained by deleting finite edges in lattice graphs while maintaining connectivity, then there exists a threshold $α_G\in[0,\infty)$ such that there do not exist ground state normalized solution if $0<a<α_G$, and there exists a ground state normalized solution if $a>α_G.$ If the graph is obtained by adding finite edges $E^{'}$ to lattice graphs, we prove that there exist $E^{'}$ and $a_1$ such that for all $a>a_1,$ there do not exist ground state normalized solutions. The second problem concerns the existence of an extremal function for the Sobolev inequality. If the graph $G$ is obtained by deleting finite edges in lattice graphs while maintaining connectivity, for the Sobolev super-critical regime, we prove that there exists an extremal function. for the Sobolev critical regime, we prove that there exists $G$ such that extremal can be attained. If the graph is obtained by adding finite edges $E^{'}$ to lattice graphs, we prove that there exists $E^{'}$ such that there does not exist an extremal function. |
| title | Constrained variational problems on perturbed lattice graphs |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.13978 |