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Stable standing waves for Nonlinear Schrödinger-Poisson system with a doping profile

Mathieu Colin, Tatsuya Watanabe

TL;DR

This work analyzes the nonlinear Schrödinger-Poisson system in $\\mathbb{R}^3$ with a doping profile $\\rho$, aiming to obtain stable standing waves as $L^2$-constraint minimizers of the energy. The authors reduce the coupled system to a single equation via $\\phi = e S(u)$ and decompose the energy into local and nonlocal terms, including $A_1(u)$, $A_2(u)$, and a constant $A_0$, with $S_1(u) \ge 0$. A scaling argument inspired by $\\cite{ZZou}$ is developed to establish strict sub-additivity of the constrained energy even in the presence of $\\rho$, yielding existence of minimizers for large mass when $\\|\\rho\\|_{\frac{6}{5}}$ and $\\|x\\cdot\\nabla\\rho\\|_{\frac{6}{5}}$ are small; this yields orbital stability of standing waves. The paper also treats the case where $\\rho$ is a characteristic function of a bounded domain by leveraging a sharp boundary trace inequality and a small geometric size condition, and proves global well-posedness in the $L^2$-subcritical regime. An open question concerns possible nonexistence of minimizers for large $\\|\\rho\\|_{\frac{6}{5}}$ in 3D.

Abstract

This paper is devoted to the study of the nonlinear Schrödinger-Poisson system with a doping profile. We are interested in the existence of stable standing waves by considering the associated $L^2$-minimization problem. The presence of a doping profile causes a difficulty in the proof of the strict sub-additivity. A key ingredient is to establish the strict sub-additivity by adapting a scaling argument, which is inspired by \cite{ZZou}. When the doping profile is a characteristic function supported on a bounded smooth domain, smallness of some geometric quantity related to the domain ensures the existence of stable standing waves.

Stable standing waves for Nonlinear Schrödinger-Poisson system with a doping profile

TL;DR

This work analyzes the nonlinear Schrödinger-Poisson system in with a doping profile , aiming to obtain stable standing waves as -constraint minimizers of the energy. The authors reduce the coupled system to a single equation via and decompose the energy into local and nonlocal terms, including , , and a constant , with . A scaling argument inspired by is developed to establish strict sub-additivity of the constrained energy even in the presence of , yielding existence of minimizers for large mass when and are small; this yields orbital stability of standing waves. The paper also treats the case where is a characteristic function of a bounded domain by leveraging a sharp boundary trace inequality and a small geometric size condition, and proves global well-posedness in the -subcritical regime. An open question concerns possible nonexistence of minimizers for large in 3D.

Abstract

This paper is devoted to the study of the nonlinear Schrödinger-Poisson system with a doping profile. We are interested in the existence of stable standing waves by considering the associated -minimization problem. The presence of a doping profile causes a difficulty in the proof of the strict sub-additivity. A key ingredient is to establish the strict sub-additivity by adapting a scaling argument, which is inspired by \cite{ZZou}. When the doping profile is a characteristic function supported on a bounded smooth domain, smallness of some geometric quantity related to the domain ensures the existence of stable standing waves.
Paper Structure (13 sections, 22 theorems, 221 equations)

This paper contains 13 sections, 22 theorems, 221 equations.

Table of Contents

  1. Introduction
  2. Variational formulation and preliminaries
  3. Reduction to a single equation
  4. Decomposition of the energy
  5. Estimates of nonlocal terms
  6. Scaling properties
  7. Nehari and Pohozaev identities
  8. Decay of $S_2(x)$ and $A_2$
  9. Existence of a minimizer
  10. Global well-posedness of the Cauchy problem
  11. Stability of standing waves
  12. The case $\rho$ is a characteristic function
  13. Concluding remark and an open problem

Key Result

Theorem 1.1

Assume eq:1.7-ASS and let $e>0$ be fixed. Suppose that $2<p<\frac{7}{3}$ and let $\mu > 2 \cdot 2^{\frac{1}{2p-4}} \mu^*$ be given. Then there exists $\rho_0= \rho_0(e, \mu)>0$ such that if $\|\rho\|_{L^{\frac{6}{5}}(\mathbb{R}^3)} +\|x \cdot \nabla \rho\|_{L^{\frac{6}{5}}(\mathbb{R}^3)} \le \rho_0$

Theorems & Definitions (44)

  • Theorem 1.1: Existence of a minimizer
  • Theorem 1.2: Orbital stability of standing wave
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 34 more