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Non-uniqueness of normalized NLS ground states on polygons with homogeneous Neumann boundary conditions

Simone Dovetta, Enrico Serra, Lorenzo Tentarelli

TL;DR

This work analyzes normalized ground states for the nonlinear Schrödinger equation on planar polygons under Neumann boundary conditions. It shows that, for powers just below the L^2-critical exponent (i.e., p in (4−ε,4)), there exists a mass μ_p for which energy ground states are not unique: two distinct minimizers share the same energy but arise from different Lagrange multipliers λ. The authors exploit the link between energy and action ground states on the Nehari manifold and perform a detailed study of L^2-critical behavior on sectors, including asymptotics as the frequency λ grows and mass thresholds dictated by the smallest interior angle of the polygon. They also establish the existence and characteristics of L^2-critical energy ground states on polygons, showing a sharp mass cutoff at μ̄_ᾱ tied to the geometry of the domain. The results reveal a non-uniqueness phenomenon under Neumann conditions and highlight differences with Dirichlet or smooth-domain cases, where the behavior can be markedly different.

Abstract

We provide a non-uniqueness result for normalized ground states of nonlinear Schrödinger equations with pure power nonlinearity on polygons with homogeneous Neumann boundary conditions, defined as global minimizers of the associated energy functional among functions with prescribed mass. Precisely, for nonlinearity powers slightly smaller than the $L^2$-critical exponent, we prove that there always exists at least one value of the mass for which normalized ground states are not unique.

Non-uniqueness of normalized NLS ground states on polygons with homogeneous Neumann boundary conditions

TL;DR

This work analyzes normalized ground states for the nonlinear Schrödinger equation on planar polygons under Neumann boundary conditions. It shows that, for powers just below the L^2-critical exponent (i.e., p in (4−ε,4)), there exists a mass μ_p for which energy ground states are not unique: two distinct minimizers share the same energy but arise from different Lagrange multipliers λ. The authors exploit the link between energy and action ground states on the Nehari manifold and perform a detailed study of L^2-critical behavior on sectors, including asymptotics as the frequency λ grows and mass thresholds dictated by the smallest interior angle of the polygon. They also establish the existence and characteristics of L^2-critical energy ground states on polygons, showing a sharp mass cutoff at μ̄_ᾱ tied to the geometry of the domain. The results reveal a non-uniqueness phenomenon under Neumann conditions and highlight differences with Dirichlet or smooth-domain cases, where the behavior can be markedly different.

Abstract

We provide a non-uniqueness result for normalized ground states of nonlinear Schrödinger equations with pure power nonlinearity on polygons with homogeneous Neumann boundary conditions, defined as global minimizers of the associated energy functional among functions with prescribed mass. Precisely, for nonlinearity powers slightly smaller than the -critical exponent, we prove that there always exists at least one value of the mass for which normalized ground states are not unique.

Paper Structure

This paper contains 8 sections, 16 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Preliminaries
  4. Action ground states on polygons
  5. Existence and qualitative properties
  6. Asymptotics for large $\lambda$ in the $L^2$-critical case
  7. $L^2$-critical energy ground states on polygons
  8. Proof of Theorem \ref{['thm-main']}

Key Result

Theorem 1.1

Let $\Omega\subset{\mathbb R}^2$ be a polygon. Then, there exists $\varepsilon>0$ (depending on $\Omega$) such that, for every $p\in(4-\varepsilon,4)$, there exists $\mu_p>0$ and $u_1,\,u_2\in H_{\mu_p}^1(\Omega)$ such that namely energy ground states of $E_p$ in $H_{\mu_p}^1(\Omega)$ are not unique.

Theorems & Definitions (32)

  • Theorem 1.1
  • Lemma 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof : Proof of Proposition \ref{['prop:AGSsector']}
  • proof : Proof of Proposition \ref{['prop:EGSsector']}
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 22 more