Stability of standing waves for theL²-critical Hartree equations with harmonic potential

Abstract

This article is concerned with the Hartree equations with harmonic potential. By an elaborate mathematical analysis, we obtain a sharp stability threshold of this equation. Then with this threshold, we prove that the standing wave of this equation exists and is stable.

Nonlinear Hartree equation in high energy-mass

“…However, in spite of quite a number of contributions dealing with this problem (see [10,11,12,13] as well as the other relevant references), almost all of them still need to restrict the energy-mass functional(or some other functionals)to be less than its minimum in one subset.…”

Nonlinear Schrödinger equations with trapping potentials in higher dimensions

“…Similar tools from functional analysis give also results regarding the stability of the stationary solutions. Then it turns out that they are stable if d = 3 or d = 4, but for d = 5 there exists a boundary frequency ω 0 dividing stable and unstable positive stationary states [23,42,65]. Going beyond the stationary solutions, in d ≤ 5 any function from Σ poses good initial data to the Cauchy problem for SNH equation [26,119].…”

“…It belongs to the wide class of nonlinear Schrödinger equations (in short NLS) and its main features are nonlocality of the nonlinearity (due to the integral, the value of the last term depends not only on the value of ψ in the given point, but also on its values in other points) and the presence of the trapping potential, specifically harmonic potential. Equations with such nonlinearity can be found in the literature under different names, such as Schrödinger-Newton [6,29,59,60,88], Schrödinger-Poisson [66,80,94], Hartree [22,23,25,42,50,65,119],…”

Orbital stability of standing waves for fractional Hartree equation with unbounded potentials

“…Remark 4.5. The case s = 1, γ = 2 (mass-critical) was studied in [14], where the threshold for the stability of standing waves for (1.1) are obtained. Earlier result on the l.w.p.…”

“…The main difficulty is the lack of proper dispersive or Strichartz estimates because the fractional Laplacian for 0 < s < 1 does not hold a control over the harmonic potential, which is shown in the deformed trajectories for the associated hamiltonian, see [15] and [19]. This is in sharp contrast to the classical NLS (s = 1) with a harmonic potential, where the existence and stability problem has been studied quite extensively [9,10,18,22,23,14,6]. Heuristically the Laplacian −∆ and V = |x| 2 have balanced strength or effect so the L 1 → L ∞ time decay t −N/2 holds locally for e it(∆−V ) .…”