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Normalized ground states for the mass supercritical Schrödinger-Bopp-Podolsky system: existence, uniqueness, limit behavior, strong instability

Juan Huang, Sheng Wang

TL;DR

This work analyzes normalized ground states for the Schrödinger–Bopp–Podolsky system with a nonlocal term and a mass-supercritical nonlinearity $p\in(\tfrac{10}{3},6)$. It establishes existence via mountain-pass geometry on the $L^2$-sphere for small mass, and derives precise concentration behavior showing convergence to the classical ground state $Q$ under suitable scalings as $m\to0^+$ or $m\to\infty$. It further proves radial symmetry and uniqueness for broad $p$-ranges through an implicit-function approach around a radial profile $W$, and shows strong instability of mountain-pass standing waves, together with a global existence threshold tied to the mountain-pass level. These results illuminate the interplay between the defocusing nonlocal term and the focusing mass-supercritical nonlinearity in a nonlocal Schrödinger model and provide a detailed picture of the asymptotic and dynamical properties of normalized states.

Abstract

This paper concerns the normalized ground states for the nonlinear Schrödinger equation in the Bopp-Podolsky electrodynamics. This equation has a nonlocal nonlinearity and a mass supercritical power nonlinearity, both of which have deep impact on the geometry of the corresponding functional, and thus on the existence, limit behavior and stability of the normalized ground states. In the present study, the existence of critical points is obtained by a mountain-pass argument developed on the $L^2$-spheres. To be specific, we show that normalized ground states exist, provided that spherical radius of the $L^2$-spheres is sufficiently small. Then, by discussing the relation between the normalized ground states of the Schrödinger-Bopp-Podolsky system and the classical Schrödinger equation, we show a precise description of the asymptotic behavior of the normalized ground states as the mass vanishes or tends to infinity. Moreover, we discuss the radial symmetry and uniqueness of the normalized ground states. Finally, the strong instability of standing waves at the mountain-pass energy level is studied by constructing an equivalent minimizing problem. Also, as a byproduct, we prove that the mountain-pass energy level gives a threshold for global existence based on this equivalent minimizing problem.

Normalized ground states for the mass supercritical Schrödinger-Bopp-Podolsky system: existence, uniqueness, limit behavior, strong instability

TL;DR

This work analyzes normalized ground states for the Schrödinger–Bopp–Podolsky system with a nonlocal term and a mass-supercritical nonlinearity . It establishes existence via mountain-pass geometry on the -sphere for small mass, and derives precise concentration behavior showing convergence to the classical ground state under suitable scalings as or . It further proves radial symmetry and uniqueness for broad -ranges through an implicit-function approach around a radial profile , and shows strong instability of mountain-pass standing waves, together with a global existence threshold tied to the mountain-pass level. These results illuminate the interplay between the defocusing nonlocal term and the focusing mass-supercritical nonlinearity in a nonlocal Schrödinger model and provide a detailed picture of the asymptotic and dynamical properties of normalized states.

Abstract

This paper concerns the normalized ground states for the nonlinear Schrödinger equation in the Bopp-Podolsky electrodynamics. This equation has a nonlocal nonlinearity and a mass supercritical power nonlinearity, both of which have deep impact on the geometry of the corresponding functional, and thus on the existence, limit behavior and stability of the normalized ground states. In the present study, the existence of critical points is obtained by a mountain-pass argument developed on the -spheres. To be specific, we show that normalized ground states exist, provided that spherical radius of the -spheres is sufficiently small. Then, by discussing the relation between the normalized ground states of the Schrödinger-Bopp-Podolsky system and the classical Schrödinger equation, we show a precise description of the asymptotic behavior of the normalized ground states as the mass vanishes or tends to infinity. Moreover, we discuss the radial symmetry and uniqueness of the normalized ground states. Finally, the strong instability of standing waves at the mountain-pass energy level is studied by constructing an equivalent minimizing problem. Also, as a byproduct, we prove that the mountain-pass energy level gives a threshold for global existence based on this equivalent minimizing problem.
Paper Structure (8 sections, 56 theorems, 358 equations)

This paper contains 8 sections, 56 theorems, 358 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The mountain-pass geometry on the constraint
  4. Compactness of the Palais-Smale sequence
  5. The behaviour of the function $m\mapsto\gamma(m)$
  6. Concentration behavior of normalized solutions
  7. The radial symmetry and uniqueness
  8. Strong instability and global existence

Key Result

Theorem 1.1

Suppose $p\in(\frac{10}{3}, 6)$ and $m>0$, then $E(u)$ has a mountain-pass geometry on $S(m)$. Moreover, there exists $m_0>0$ such that for any $m\in(0, m_0)$, Eq.(main5) has a couple of weak solution $(u_m, \omega_m)\in H^1(\mathbb{R}^3)\times\mathbb{R}^+$ with $||u_m||_{L^2(\mathbb{R}^3)}=m$ and $

Theorems & Definitions (101)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1: Local well-posedness, Zheng2022
  • Lemma 2.1: Hardy-Littlewood-Sobolev inequality, Lieb2001
  • Corollary 2.1
  • Lemma 2.2: dAvenia2019
  • ...and 91 more