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Non-Kirchhoff Vertices and NLS Ground States on graphs

Riccardo Adami, Filippo Boni, Alice Ruighi

Abstract

We review some recent results and announce some new ones on the problem of the existence of ground states for the Nonlinear Schrödinger Equation on graphs endowed with vertices where the matching condition, instead of being free (or Kirchhoff's), is non-trivially interacting. In this category fall Dirac's delta conditions, delta prime, Fülöp-Tsutsui, and others.

Non-Kirchhoff Vertices and NLS Ground States on graphs

Abstract

We review some recent results and announce some new ones on the problem of the existence of ground states for the Nonlinear Schrödinger Equation on graphs endowed with vertices where the matching condition, instead of being free (or Kirchhoff's), is non-trivially interacting. In this category fall Dirac's delta conditions, delta prime, Fülöp-Tsutsui, and others.

Paper Structure

This paper contains 8 sections, 9 theorems, 26 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Linear non-Kirchhoff's condition
  3. Minimization of the energy under the mass constraint
  4. Minimization of the action under the Nehari's constraint
  5. Nonlinear delta conditions
  6. Subcritical case
  7. Critical cases
  8. Conclusion and perspectives

Key Result

Theorem 2.1

Let $\alpha >0$. Then there exists a unique nonnegative minimizer $\phi_{\omega}$ of Somega under the Nehari's constraint. Moreover,

Figures (2)

  • Figure 1: ${\mathcal{G}}=(\mathcal{V},{\mathcal{E}})$
  • Figure 2: 3-tail state

Theorems & Definitions (9)

  • Theorem 2.1: Attractive case, Proposition 2 and Theorem 1 in foo
  • Theorem 2.2: Repulsive case, Theorem 1 and 2 in fj
  • Theorem 2.3: Theorem 5.3, Proposition 6.11 and Theorem 6.13 in an
  • Theorem 2.4
  • Proposition 2.5
  • Theorem 3.1: Theorem 1.3 in bd
  • Theorem 3.2
  • Theorem 3.3: Theorem 1.4 in bd
  • Theorem 3.4: Theorem 1.5 in bd