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Normalized ground states for the NLS equation with combined nonlinearities

Nicola Soave

TL;DR

This work analyzes ground states for the nonlinear Schrödinger equation with combined nonlinearities under a mass constraint, revealing how the relative strength and type of the two nonlinearities (subcritical, critical, supercritical) shapes the energy geometry and dynamics. Through a variational framework on the mass shell, Pohozaev manifolds, and fiber maps that preserve the $L^2$-norm, the authors establish existence and stability results for ground states across regimes, and derive detailed asymptotics as the lower-order perturbation parameter $\mu$ vanishes or as the subcritical exponent approaches the $L^2$-critical value. They show the emergence of multiple stationary states (local minimizers and mountain-pass solutions) in certain regimes, with stability for small $\mu$ and strong instability in others, and they connect these static properties to the dynamics by providing global existence versus finite-time blow-up criteria for the time-dependent NLS. The results illuminate how a focusing or defocusing lower-order term can dramatically alter the variational landscape, leading to discontinuities in ground-state energy and rich bifurcation behavior that inform the long-time dynamics of solutions. Overall, the paper advances understanding of normalized ground states in mixed-power NLS and offers precise asymptotics and stability criteria that are relevant for applications in nonlinear optics and Bose-Einstein condensates.

Abstract

We study existence and properties of ground states for the nonlinear Schrödinger equation with combined power nonlinearities \[ -Δu= λu + μ|u|^{q-2} u + |u|^{p-2} u \qquad \text{in $\mathbb{R}^N$, $N \ge 1$,} \] having prescribed mass \[ \int_{\mathbb{R}^N} |u|^2 = a^2. \] Under different assumptions on $q<p$, $a>0$ and $μ\in \mathbb{R}$ we prove several existence and stability/instability results. In particular, we consider cases when \[ 2<q \le 2+ \frac{4}{N} \le p<2^*, \quad q \neq p, \] i.e. the two nonlinearities have different character with respect to the $L^2$-critical exponent. These cases present substantial differences with respect to purely subcritical or supercritical situations, which were already studied in the literature. We also give new criteria for global existence and finite time blow-up in the associated dispersive equation.

Normalized ground states for the NLS equation with combined nonlinearities

TL;DR

This work analyzes ground states for the nonlinear Schrödinger equation with combined nonlinearities under a mass constraint, revealing how the relative strength and type of the two nonlinearities (subcritical, critical, supercritical) shapes the energy geometry and dynamics. Through a variational framework on the mass shell, Pohozaev manifolds, and fiber maps that preserve the -norm, the authors establish existence and stability results for ground states across regimes, and derive detailed asymptotics as the lower-order perturbation parameter vanishes or as the subcritical exponent approaches the -critical value. They show the emergence of multiple stationary states (local minimizers and mountain-pass solutions) in certain regimes, with stability for small and strong instability in others, and they connect these static properties to the dynamics by providing global existence versus finite-time blow-up criteria for the time-dependent NLS. The results illuminate how a focusing or defocusing lower-order term can dramatically alter the variational landscape, leading to discontinuities in ground-state energy and rich bifurcation behavior that inform the long-time dynamics of solutions. Overall, the paper advances understanding of normalized ground states in mixed-power NLS and offers precise asymptotics and stability criteria that are relevant for applications in nonlinear optics and Bose-Einstein condensates.

Abstract

We study existence and properties of ground states for the nonlinear Schrödinger equation with combined power nonlinearities having prescribed mass Under different assumptions on , and we prove several existence and stability/instability results. In particular, we consider cases when i.e. the two nonlinearities have different character with respect to the -critical exponent. These cases present substantial differences with respect to purely subcritical or supercritical situations, which were already studied in the literature. We also give new criteria for global existence and finite time blow-up in the associated dispersive equation.

Paper Structure

This paper contains 16 sections, 43 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Acknowledgments
  4. Preliminaries
  5. $L^2$-critical leading term
  6. Compactness of Palais-Smale sequences in the $L^2$-supercritical setting
  7. Supercritical leading term with focusing subcritical perturbation
  8. Supercritical leading term with focusing critical perturbation
  9. Supercritical leading term with defocusing perturbation
  10. Properties of ground states I
  11. Description of $Z_{a,\mu}$
  12. Strong instability of the $e^{-i \hat{\lambda} t} \hat{u}$
  13. Asymptotic behavior as $\mu \to 0^+$: proof of Theorem \ref{['thm: mu to 0']}
  14. Asymptotic behavior as $q \to \bar{p}^-$: proof of Theorem \ref{['thm: q to bar p']}
  15. Properties of ground states II
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $N \ge 1$, $2<q<p=\bar{p}$. It results that:

Theorems & Definitions (101)

  • Definition 1
  • Definition 2
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.1
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.2
  • ...and 91 more