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Multiple normalized solutions to a system of nonlinear Schrödinger equations

Jarosław Mederski, Andrzej Szulkin

TL;DR

This work investigates normalized solutions to a coupled system of nonlinear Schrödinger equations in ${\mathbb R}^3$ with prescribed $L^2$-norms. By formulating a constrained variational problem on the radial subspace and employing a Nehari–Pohožaev constraint, the authors establish a robust Palais–Smale framework aided by the Cwikel–Lieb–Rozenblum theorem to bound Morse indices, along with a Liouville-type exterior-domain lemma. They prove multiplicity results for the case $K=2$ under large positive couplings, and a general existence result for $K\ge 2$ under a coupling condition, plus a nonexistence result for ground states in the repulsive coupling regime. The novel combination of Morse-index estimates via CLR and exterior-domain Liouville arguments advances the understanding of normalized solutions in multi-component NLS systems and provides concrete multiplicity and nonexistence criteria with implications for Bose–Einstein condensates and nonlinear optics.

Abstract

We find a normalized solution $u=(u_1,\ldots,u_K)$ to the system of $K$ coupled nonlinear Schrödinger equations \begin{equation*} \left\{ \begin{array}{l} -Δu_i+ λ_i u_i = \sum_{j=1}^Kβ_{i,j}u_i|u_i|^{p/2-2}|u_j|^{p/2} \quad \mathrm{in} \, \mathbb{R}^3,\newline u_i \in H^1_{rad}(\mathbb{R}^3),\newline \int_{\mathbb{R}^3} |u_i|^2 \, dx = ρ_i^2 \quad \text{for }i=1,\ldots, K, \end{array} \right. \end{equation*} where $ρ=(ρ_1,\ldots,ρ_K)\in(0,\infty)^K$ is prescribed, $(λ,u) \in \mathbb{R}^K\times H^1(\mathbb{R}^3)^K$ are the unknown and $4\leq p<6$. In the case of two equations we show the existence of multiple solutions provided that the coupling is sufficiently large. We also show that for negative coupling there are no ground state solutions. The main novelty in our approach is that we use the Cwikel-Lieb-Rozenblum theorem in order to estimate the Morse index of a solution as well as a Liouville-type result in an exterior domain.

Multiple normalized solutions to a system of nonlinear Schrödinger equations

TL;DR

This work investigates normalized solutions to a coupled system of nonlinear Schrödinger equations in with prescribed -norms. By formulating a constrained variational problem on the radial subspace and employing a Nehari–Pohožaev constraint, the authors establish a robust Palais–Smale framework aided by the Cwikel–Lieb–Rozenblum theorem to bound Morse indices, along with a Liouville-type exterior-domain lemma. They prove multiplicity results for the case under large positive couplings, and a general existence result for under a coupling condition, plus a nonexistence result for ground states in the repulsive coupling regime. The novel combination of Morse-index estimates via CLR and exterior-domain Liouville arguments advances the understanding of normalized solutions in multi-component NLS systems and provides concrete multiplicity and nonexistence criteria with implications for Bose–Einstein condensates and nonlinear optics.

Abstract

We find a normalized solution to the system of coupled nonlinear Schrödinger equations \begin{equation*} \left\{ \begin{array}{l} -Δu_i+ λ_i u_i = \sum_{j=1}^Kβ_{i,j}u_i|u_i|^{p/2-2}|u_j|^{p/2} \quad \mathrm{in} \, \mathbb{R}^3,\newline u_i \in H^1_{rad}(\mathbb{R}^3),\newline \int_{\mathbb{R}^3} |u_i|^2 \, dx = ρ_i^2 \quad \text{for }i=1,\ldots, K, \end{array} \right. \end{equation*} where is prescribed, are the unknown and . In the case of two equations we show the existence of multiple solutions provided that the coupling is sufficiently large. We also show that for negative coupling there are no ground state solutions. The main novelty in our approach is that we use the Cwikel-Lieb-Rozenblum theorem in order to estimate the Morse index of a solution as well as a Liouville-type result in an exterior domain.
Paper Structure (6 sections, 12 theorems, 95 equations)

This paper contains 6 sections, 12 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['th:PS']}
  3. Proof of Theorem \ref{['th:mainK2']}
  4. Proof of Theorem \ref{['th:mainK']}
  5. Nonexistence of ground states
  6. A Liouville-type lemma

Key Result

Theorem 1.1

Let $4\leq p<6$ and suppose that $K=2$. For any $m\geq 1$ there is a constant $\beta_m>0$ such that if $\beta := \beta_{1,2} = \beta_{2,1}>\beta_m$, then eq either has infinitely many solutions or at least $m$ solutions $u^j\in H^1_{rad}(\mathbb{R}^3)^2$ such that $J(u^1)<J(u^2)<\cdots<J(u^m)$. One

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • proof
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 2.1
  • Lemma 3.1: MederskiSchinoJeanjeanLuNorm
  • Proposition 3.2
  • ...and 12 more