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Multiple positive bound states for NLS equations on noncompact metric graphs with an attractive potential

Q. Liu

TL;DR

The paper addresses the existence and multiplicity of bound states for the subcritical NLS on a noncompact metric graph $G$ with an attractive potential $W$, under a fixed mass constraint $\|u\|_{L^2(G)}^2=\mu$. It develops a doubly constrained variational framework, minimizing the energy $E(u,G)$ on the mass shell while enforcing an edge-localization constraint, and uses concentration-compactness adapted to graphs to obtain minimizers for large $\mu$. The main result is that if $G$ has $k$ bounded edges, then there exist at least $k$ geometrically distinct bound states, each attaining its maximum on a different bounded edge; these are not necessarily ground states. This work highlights how graph topology governs the multiplicity of localized standing waves in networks with attractive potentials and provides a constructive approach to generate multiple bound states.

Abstract

In this paper, we establish the existence of bounded states and geometrically distinct solutions for the subcritical NLS equation with attractive potential on metric graphs $\mathcal{G}$ when the mass $μ$ is large enough.We show that the NLS equation exists at least as many bound states of mass $μ$ as the number of bounded edges of $\mathcal{G}$ if the attractive potential satisfies some suitable assumptions.It is worth noting that this is different from the case of ground state, which on some graphs may fail to exist for every value of $μ$.

Multiple positive bound states for NLS equations on noncompact metric graphs with an attractive potential

TL;DR

The paper addresses the existence and multiplicity of bound states for the subcritical NLS on a noncompact metric graph with an attractive potential , under a fixed mass constraint . It develops a doubly constrained variational framework, minimizing the energy on the mass shell while enforcing an edge-localization constraint, and uses concentration-compactness adapted to graphs to obtain minimizers for large . The main result is that if has bounded edges, then there exist at least geometrically distinct bound states, each attaining its maximum on a different bounded edge; these are not necessarily ground states. This work highlights how graph topology governs the multiplicity of localized standing waves in networks with attractive potentials and provides a constructive approach to generate multiple bound states.

Abstract

In this paper, we establish the existence of bounded states and geometrically distinct solutions for the subcritical NLS equation with attractive potential on metric graphs when the mass is large enough.We show that the NLS equation exists at least as many bound states of mass as the number of bounded edges of if the attractive potential satisfies some suitable assumptions.It is worth noting that this is different from the case of ground state, which on some graphs may fail to exist for every value of .
Paper Structure (3 sections, 10 theorems, 114 equations, 5 figures)

This paper contains 3 sections, 10 theorems, 114 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of theorem \ref{['thm1']}

Key Result

Theorem 1.2

Let $\mathcal{G}$ be a noncompact and connected metric graph, having $k$ bounded edges $\{e_i\}_{i=1}^k$. Then equation eq1 admits at least $k$ bound states $\{u_i\}_{i=1}^k$ for large $\mu$. Moreover, the maximum of $u_i$ is attained only at the edge $e_i$ and either $u_i>0$ or $u_i<0$.

Figures (5)

  • Figure 1: A graph with 7 vertices and 14 edges: 3 unbounded edges (halflines) and 11 bounded edges.
  • Figure 2: $\mathrm{card}\{x\in \mathcal{G}:\; u(x)=t \}\geq 2$
  • Figure 3: $\mathrm{card}\{x\in \mathcal{G}:\; u(x)=t \}\geq 3$
  • Figure 4: $\mathrm{card}\{x\in \mathcal{G}:\; u(x)=t \}\geq 3$
  • Figure 5: The graph $\mathcal{G}^\prime$ generated from $\mathcal{G}$

Theorems & Definitions (19)

  • Remark 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 9 more