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Constrained variational problems on perturbed lattice graphs

Weiqi Guan

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 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.

Constrained variational problems on perturbed lattice graphs

Abstract

In this paper, we solve some constrained variational problems on perturbed lattice graphs . 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 such that there do not exist ground state normalized solution if , and there exists a ground state normalized solution if If the graph is obtained by adding finite edges to lattice graphs, we prove that there exist and such that for all 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 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 such that extremal can be attained. If the graph is obtained by adding finite edges to lattice graphs, we prove that there exists such that there does not exist an extremal function.
Paper Structure (4 sections, 21 theorems, 50 equations)

This paper contains 4 sections, 21 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Perturbation by deleting edges
  4. perturbation by adding edges

Key Result

Theorem 1.1

Let $G=(V,E\backslash {E}^{'})$ be a connected graph obtained by deleting finite edges $\mathcal{E}^{'}$ in lattice graphs $\mathbb{Z}^d$ but keep the connectivity. Then we have (1)If $d\geq 1$ and $p>2$, then there exists a threshold $\alpha_G$ such that $E_{G}^{a,p}$ can be attained if $0< a <\alp

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Corollary 2.4
  • proof
  • Lemma 2.5
  • ...and 26 more