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Minimizers of mass-constrained functionals involving a nonattractive point interaction

Gustavo de Paula Ramos

TL;DR

This work proves the existence of mass-constrained minimizers for a class of functionals involving a nonattractive point interaction in $\mathbb{R}^3$ by establishing compactness of minimizing sequences after ruling out vanishing and dichotomy. The authors construct and employ the energy space $W^{1,2}_\alpha$ associated with the point interaction $-\Delta_\alpha$, and develop an abstract minimization framework $I_\alpha$ together with an auxiliary problem to study coercivity and concentration-compactness phenomena. They adapt strategies from Adami et al and Bellazzini & Siciliano to show nonvanishing and strict subadditivity, yielding existence of ground states for small mass in two nonlinear models: a Kirchhoff-type equation and a Schrödinger–Poisson system with a point interaction. The results elucidate the influence of the point defect on variational structure and provide rigorous ground-state existence under small-mass regimes, highlighting the analytic role of the Green functions and energy-splitting techniques in three dimensions.

Abstract

We establish conditions to ensure the existence of minimizer for a class of mass-constrained functionals involving a nonattractive point interaction in three dimensions. The existence of minimizers follows from the compactness of minimizing sequences which holds when we can simultaneously rule out the possibilities of vanishing and dichotomy. The proposed method is derived from the strategy used to avoid vanishing in Adami, Boni, Carlone & Tentarelli (Calc. Var. 61, 195 (2022)) and the strategy used to avoid dichotomy in Bellazzini & Siciliano (J. Funct. Anal. 261, 9 (2011)). As applications, we prove the existence of ground states with sufficiently small mass for the following nonlinear problems with a point interaction: a Kirchhoff-type equation and the Schrödinger-Poisson system.

Minimizers of mass-constrained functionals involving a nonattractive point interaction

TL;DR

This work proves the existence of mass-constrained minimizers for a class of functionals involving a nonattractive point interaction in by establishing compactness of minimizing sequences after ruling out vanishing and dichotomy. The authors construct and employ the energy space associated with the point interaction , and develop an abstract minimization framework together with an auxiliary problem to study coercivity and concentration-compactness phenomena. They adapt strategies from Adami et al and Bellazzini & Siciliano to show nonvanishing and strict subadditivity, yielding existence of ground states for small mass in two nonlinear models: a Kirchhoff-type equation and a Schrödinger–Poisson system with a point interaction. The results elucidate the influence of the point defect on variational structure and provide rigorous ground-state existence under small-mass regimes, highlighting the analytic role of the Green functions and energy-splitting techniques in three dimensions.

Abstract

We establish conditions to ensure the existence of minimizer for a class of mass-constrained functionals involving a nonattractive point interaction in three dimensions. The existence of minimizers follows from the compactness of minimizing sequences which holds when we can simultaneously rule out the possibilities of vanishing and dichotomy. The proposed method is derived from the strategy used to avoid vanishing in Adami, Boni, Carlone & Tentarelli (Calc. Var. 61, 195 (2022)) and the strategy used to avoid dichotomy in Bellazzini & Siciliano (J. Funct. Anal. 261, 9 (2011)). As applications, we prove the existence of ground states with sufficiently small mass for the following nonlinear problems with a point interaction: a Kirchhoff-type equation and the Schrödinger-Poisson system.
Paper Structure (29 sections, 32 theorems, 189 equations)

This paper contains 29 sections, 32 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Motivation and goals
  3. Notation
  4. The operator $- \Delta_\alpha$
  5. The energy space $W^{1, 2}_\alpha$
  6. Considered problem
  7. The abstract minimization problem
  8. An auxiliary minimization problem
  9. Existence of solution to \ref{['eqn:AMP']}
  10. How may compactness fail?
  11. Avoiding vanishing
  12. Avoiding dichotomy
  13. Applications
  14. A Kirchhoff-type equation involving a point interaction
  15. The Schrödinger--Poisson system involving a point interaction
  16. ...and 14 more sections

Key Result

Corollary 1.1

Suppose that $\alpha \in \mathbb{R}$; $\lambda, \mu \in \rbrack0, \infty\lbrack$ and $q \in \mathbb{C}$.

Theorems & Definitions (57)

  • Corollary 1.1
  • Lemma 1.2
  • Definition 1.3
  • Lemma 1.4
  • Lemma 1.5
  • Lemma 1.6
  • Remark 1.7
  • Definition 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 47 more