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Uniqueness of normalized ground states for NLS models

Hichem Hajaiej, Linjie Song

TL;DR

This work addresses the uniqueness of normalized ground states for nonlinear Schrödinger-type equations by developing two general frameworks. The first method uses a sign-change analysis of the λ-derivative along a global solution branch to obtain strict monotonicity of the $L^2$ mass and, hence, ground-state uniqueness on each mass level; the second method connects the differentiability of the minimal energy $m(c)$ on the mass constraint set $S_c$ with the uniqueness of ground states via the set of corresponding Lagrange multipliers. The authors apply these approaches to a broad class of problems, including with or without potentials, fractional operators, and inhomogeneous nonlinearities on the unit ball, deriving existence, mass-monotonicity, uniqueness, and orbital stability results across mass-subcritical, critical, and supercritical regimes. The results unify and extend known outcomes, providing conditions under which normalized ground states are unique and orbitally stable, with explicit constructions of global solution branches. The work thus advances the understanding of how spectral and variational structures govern standing-wave stability in nonlinear dispersive PDEs. $

Abstract

We present two methods to prove the uniqueness of normalized ground states. We will first discuss the key ideas and ingredients of each method. Then, we will apply them to various classes of PDEs. Our approach is applicable to other operators, domains and nonlinearities provided that some hypotheses are satisfied.

Uniqueness of normalized ground states for NLS models

TL;DR

This work addresses the uniqueness of normalized ground states for nonlinear Schrödinger-type equations by developing two general frameworks. The first method uses a sign-change analysis of the λ-derivative along a global solution branch to obtain strict monotonicity of the mass and, hence, ground-state uniqueness on each mass level; the second method connects the differentiability of the minimal energy on the mass constraint set with the uniqueness of ground states via the set of corresponding Lagrange multipliers. The authors apply these approaches to a broad class of problems, including with or without potentials, fractional operators, and inhomogeneous nonlinearities on the unit ball, deriving existence, mass-monotonicity, uniqueness, and orbital stability results across mass-subcritical, critical, and supercritical regimes. The results unify and extend known outcomes, providing conditions under which normalized ground states are unique and orbitally stable, with explicit constructions of global solution branches. The work thus advances the understanding of how spectral and variational structures govern standing-wave stability in nonlinear dispersive PDEs. $

Abstract

We present two methods to prove the uniqueness of normalized ground states. We will first discuss the key ideas and ingredients of each method. Then, we will apply them to various classes of PDEs. Our approach is applicable to other operators, domains and nonlinearities provided that some hypotheses are satisfied.
Paper Structure (16 sections, 29 theorems, 145 equations)

This paper contains 16 sections, 29 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm2.10']}
  3. Hypotheses and main results for \ref{['eq1.1']} and \ref{['eq1.2']}
  4. NLS with or without a potential
  5. NLS with inhomogeneous nonlinearities on the unit ball
  6. Proofs of Theorems \ref{['thm2.1']}-\ref{['thm2.5']}
  7. Monotonicity of $L^2$ mass of solutions on the global branch for \ref{['eq1.1']}
  8. Uniqueness of the normalized ground state for \ref{['eq1.1']}
  9. Proofs of Theorems \ref{['thm2.7']}-\ref{['thm2.9']}
  10. Existence and uniqueness of the global branch and monotonicity of $L^2$ mass of solutions on this branch for \ref{['eq1.2']}
  11. Uniqueness of the normalized ground state for \ref{['eq1.2']}
  12. Orbital stability results
  13. A counter-example where the normalized ground state exists but is not unique
  14. Existence and uniqueness of the global branch
  15. The relationship between the existence/uniqueness of the normalized ground state and the existence/uniqueness of ground state solution under Nehari manifold
  16. ...and 1 more sections

Key Result

Theorem 1.1

Assume that $(h)$ holds and $2 < p < 2 + (2\theta + 4s)/N$. Then

Theorems & Definitions (68)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 2.1: Theorem 5.16 and Lemma 5.13 in HS
  • Lemma 2.2
  • proof
  • ...and 58 more