Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multiplicity result for a mass supercritical NLS with a partial confinement

Louis Jeanjean, Linjie Song

TL;DR

This work proves the existence of two distinct normalized bound states for a mass-supercritical NLS with partial confinement in ${\mathbb R}^3$: a ground state and a mountain-pass state, for small mass $\mu$. The authors devise a non-perturbative scheme by first solving the constrained problem on large balls ${B_R}$, exploiting symmetry to gain compactness, and then passing to the limit $R\to\infty$ to obtain two positive critical points of the energy on the mass sphere ${S_\mu}$. A key compactness result using symmetry and a Liouville-type argument ensures strong convergence in $H$ away from the critical energy threshold, enabling a rigorous limit passage and the construction of the two solutions on the full space. Moreover, the paper provides detailed asymptotic descriptions as $\mu\to0$, including precise behavior of the Lagrange multipliers and the profiles of the two solutions, and discusses orbital stability properties. Overall, the work extends non-perturbative multiplicity results for NLS with partial confinement and highlights the utility of bounded-domain approximation combined with symmetry-based compactness.

Abstract

We consider an NLS equation in $\mathbb{R}^3$ with partial confinement and mass supercritical nonlinearity. In Bellazzini, Boussaid, Jeanjean and Visciglia (Comm. Math. Phys. 353, 2017, 229-251) for such a problem, a solution with a prescribed $L^2$ norm was obtained, as a local minima, and the existence of a second solution, at a mountain pass energy level, was proposed as an open problem. We give here a positive, non-perturbative, answer to this problem. Our solution is obtained as a limit of a sequence of solutions of corresponding problems on bounded domains of $\mathbb{R}^3$. The symmetry of solutions on bounded domains is used centrally in the convergence process.

Multiplicity result for a mass supercritical NLS with a partial confinement

TL;DR

This work proves the existence of two distinct normalized bound states for a mass-supercritical NLS with partial confinement in : a ground state and a mountain-pass state, for small mass . The authors devise a non-perturbative scheme by first solving the constrained problem on large balls , exploiting symmetry to gain compactness, and then passing to the limit to obtain two positive critical points of the energy on the mass sphere . A key compactness result using symmetry and a Liouville-type argument ensures strong convergence in away from the critical energy threshold, enabling a rigorous limit passage and the construction of the two solutions on the full space. Moreover, the paper provides detailed asymptotic descriptions as , including precise behavior of the Lagrange multipliers and the profiles of the two solutions, and discusses orbital stability properties. Overall, the work extends non-perturbative multiplicity results for NLS with partial confinement and highlights the utility of bounded-domain approximation combined with symmetry-based compactness.

Abstract

We consider an NLS equation in with partial confinement and mass supercritical nonlinearity. In Bellazzini, Boussaid, Jeanjean and Visciglia (Comm. Math. Phys. 353, 2017, 229-251) for such a problem, a solution with a prescribed norm was obtained, as a local minima, and the existence of a second solution, at a mountain pass energy level, was proposed as an open problem. We give here a positive, non-perturbative, answer to this problem. Our solution is obtained as a limit of a sequence of solutions of corresponding problems on bounded domains of . The symmetry of solutions on bounded domains is used centrally in the convergence process.

Paper Structure

This paper contains 7 sections, 18 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problems on $B_R$
  4. Problems on ${\mathbb R}^3$
  5. A compactness result
  6. Proof of Theorem \ref{['thmmulti']}
  7. Proof of Theorem \ref{['thmasym']}

Key Result

Theorem 1.2

Let $7/3 < p <5$ and where $C_{p+1} >0$ and $\Lambda_0$ are defined in eqgninequ and infspectrum respectively. Then, for any $0 < \mu < \mu_1$, there is at least two positive critical points $u_1, u_2$ of $E$ constrained on $S_\mu$, $u_1$ is a ground state at the level $c_\mu$ and $u_2$ is at the mountain pass level $m_

Theorems & Definitions (39)

  • Definition 1.1: Ground state
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 2.1
  • proof : Proof of Proposition \ref{['propsymtocom']}
  • Lemma 2.2
  • Remark 2.3
  • ...and 29 more