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Strong instability of standing waves for $L^2$-supercritical Schrödinger-Poisson system with a doping profile

Mathieu Colin, Tatsuya Watanabe

TL;DR

The paper analyzes strong instability of standing waves for the $L^2$-supercritical Schrödinger-Poisson system with a doping profile, establishing instability of ground states under small coupling and doping effects. It develops a new energy inequality along the $L^2$-invariant scaling and adopts the Fukaya-Ohta framework to trigger finite-time blow-up via a virial identity, without requiring a fixed frequency. When the doping is a characteristic function on a bounded domain, instability is linked to geometric quantities such as mean curvature, with a sharp boundary-trace argument and moving-surface calculus enabling a smallness condition on domain geometry to guarantee instability. The results extend the instability theory for nonlocal, inhomogeneous media and demonstrate how inhomogeneities and geometry influence the dynamics of standing waves in Schrödinger-Poisson systems.

Abstract

This paper is devoted to the study of the nonlinear Schrödinger-Poisson system with a doping profile. We are interested in the strong instability of standing waves associated with ground state solutions in the $L^2$-supercritical case. The presence of a doping profile causes several difficulties, especially in examining geometric shapes of fibering maps along an $L^2$-invariant scaling curve. Furthermore, the classical approach by Berestycki-Cazenave for the strong instability cannot be applied to our problem due to a remainder term caused by the doping profile. To overcome these difficulties, we establish a new energy inequality associated with the $L^2$-invariant scaling and adopt the strong instability result developed by Fukaya-Ohta(2018). When the doping profile is a characteristic function supported on a bounded smooth domain, some geometric quantities related to the domain, such as the mean curvature, are responsible for the strong instability of standing waves.

Strong instability of standing waves for $L^2$-supercritical Schrödinger-Poisson system with a doping profile

TL;DR

The paper analyzes strong instability of standing waves for the -supercritical Schrödinger-Poisson system with a doping profile, establishing instability of ground states under small coupling and doping effects. It develops a new energy inequality along the -invariant scaling and adopts the Fukaya-Ohta framework to trigger finite-time blow-up via a virial identity, without requiring a fixed frequency. When the doping is a characteristic function on a bounded domain, instability is linked to geometric quantities such as mean curvature, with a sharp boundary-trace argument and moving-surface calculus enabling a smallness condition on domain geometry to guarantee instability. The results extend the instability theory for nonlocal, inhomogeneous media and demonstrate how inhomogeneities and geometry influence the dynamics of standing waves in Schrödinger-Poisson systems.

Abstract

This paper is devoted to the study of the nonlinear Schrödinger-Poisson system with a doping profile. We are interested in the strong instability of standing waves associated with ground state solutions in the -supercritical case. The presence of a doping profile causes several difficulties, especially in examining geometric shapes of fibering maps along an -invariant scaling curve. Furthermore, the classical approach by Berestycki-Cazenave for the strong instability cannot be applied to our problem due to a remainder term caused by the doping profile. To overcome these difficulties, we establish a new energy inequality associated with the -invariant scaling and adopt the strong instability result developed by Fukaya-Ohta(2018). When the doping profile is a characteristic function supported on a bounded smooth domain, some geometric quantities related to the domain, such as the mean curvature, are responsible for the strong instability of standing waves.

Paper Structure

This paper contains 11 sections, 24 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Well-posedness of the Cauchy problem and the virial identity
  3. Variational setting and preliminaries
  4. Decomposition of the energy
  5. Estimates of nonlocal terms
  6. Scaling properties
  7. Nehari and Pohozaev identities
  8. Properties of ground state solutions
  9. Fibering maps along $L^2$-invariant scaling
  10. Strong instability of standing waves
  11. The case $\rho$ is a characteristic function

Key Result

Proposition 1.1

Suppose that $2<p<5$ and assume that (A1)-(A2) are satisfied. There exists $\rho_0$ independent of $e$, $\rho$ such that if then eq:1.1 has a ground state solution $u_0$. Moreover any ground state solution of eq:1.1 is real-valued up to phase shift.

Theorems & Definitions (44)

  • Proposition 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • Proposition 4.1
  • Proposition 4.2
  • Lemma 4.3
  • ...and 34 more