Constrained variational problems on perturbed lattice graphs

TL;DR

This paper investigates the existence of solutions to Schrödinger equations and extremal functions for Sobolev inequalities on perturbed lattice graphs, identifying conditions under which solutions exist or do not exist based on graph modifications.

Contribution

It provides new results on the existence thresholds for ground state solutions and extremal functions on perturbed lattice graphs, considering edge deletions and additions.

Findings

Existence of a threshold _G for ground state solutions when edges are deleted.

No ground state solutions for small  when edges are deleted.

Existence of extremal functions in super-critical Sobolev regimes on certain perturbed graphs.

Abstract

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 Schr\"odinger equations \begin{equation*} \left\{ \begin{aligned} &-\Delta_{G} u+\lambda u=\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 $\alpha_G\in[0,\infty)$ such that there do not exist ground state normalized solution if $0<a<\alpha_G$, and there exists a ground state normalized solution if $a>\alpha_G.$ 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.