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Normalized solutions of nonlinear magnetic Schrödinger equations on metric graphs

Pietro d'Avenia, Zhentao He, Chao Ji

Abstract

In this paper we first establish the theory of a magnetic Sobolev space $H^1_A(\mathcal{G},\mathbb{C})$ on metric graphs $\mathcal{G}$ and we prove the self-adjointness of its corresponding magnetic Schrödinger operator. Then, in this setting, we investigate the existence and multiplicity of normalized solutions to nonlinear magnetic Schrödinger equations on compact metric graphs and on noncompact metric graphs with localized nonlinearities or nonlinearities acting on whole metric graphs, covering the mass-subcritical, mass-critical, and mass-supercritical cases.

Normalized solutions of nonlinear magnetic Schrödinger equations on metric graphs

Abstract

In this paper we first establish the theory of a magnetic Sobolev space on metric graphs and we prove the self-adjointness of its corresponding magnetic Schrödinger operator. Then, in this setting, we investigate the existence and multiplicity of normalized solutions to nonlinear magnetic Schrödinger equations on compact metric graphs and on noncompact metric graphs with localized nonlinearities or nonlinearities acting on whole metric graphs, covering the mass-subcritical, mass-critical, and mass-supercritical cases.
Paper Structure (17 sections, 24 theorems, 222 equations)

This paper contains 17 sections, 24 theorems, 222 equations.

Table of Contents

  1. Introduction
  2. Second order nonlinear equations on metric graphs
  3. Nonlinear magnetic Schrödinger equations on metric graphs
  4. The functional setting
  5. Our results
  6. Further comments and remarks
  7. Plan of the paper
  8. Magnetic Sobolev space and self-adjointness of the magnetic Schrödinger operator
  9. Preliminaries
  10. Diamagnetic inequality
  11. Gagliardo-Nirenberg-Sobolev inequalities
  12. A non-existence result
  13. Modified functionals
  14. Proofs of Theorems \ref{['th2']} and \ref{['th3']}
  15. Proof of Theorem \ref{['th4']}
  16. ...and 2 more sections

Key Result

Theorem 1.2

Let $\mathcal{G}$ be any compact metric graph or any noncompact metric graph with a finite number of edges and a non-empty compact core $\mathcal{K}$, and let $p > 2$. Then $E( \cdot,\mathcal{G})$ has infinitely many critical points $\{u_n\}\subset H^1_A(\mathcal{G},\mathbb{C})$ such that $\lim_{n \

Theorems & Definitions (56)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 46 more