Ground states for a nonlocal cubic-quartic Gross-Pitaevskii equation

Abstract

We prove existence and qualitative properties of ground state solutions to a generalized nonlocal 3rd-4th order Gross-Pitaevskii equation. Using a mountain pass argument on spheres and constructing appropriately localized Palais-Smale sequences we are able to prove existence of real positive ground states as saddle points. The analysis is deployed in the set of possible states, thus overcoming the problem that the energy is unbounded below. We also prove a corresponding nonlocal Pohozaev identity with no rest term, a crucial part of the analysis.

On 3d dipolar Bose-Einstein condensates involving quantum fluctuations and three-body interactions

“…Remark 1.3. Our main results complete the results of [21,22], which studied (1.5) with p ∈ (4,6]. Due to the dipolar term, the energy functional E(u) is not invariant under rotations and this lack of symmetry prevents us from working in the radial space H 1 rad (R 3 ).…”

“…When p = 6, (1.5) with λ 3 > 0 describes the short-range conservative three-body interactions (see [7]) and (1.5) with λ 3 < 0 models three-body losses (see [25]), respectively. For detailed physical backgrounds on (1.5) and further references, one can refer to [8,11,15,21,22,27,31,32] and the references therein. R. Carles et al [11] concerned with the existence and uniqueness of solution to…”

“…The very recent works of Y. M. Luo and A. Stylianou [21,22] were devoted to the study of (1.5) in two cases: λ 3 < 0 and p = 5; λ 3 > 0 and p ∈ (4, 6], respectively. In [21], the authors adopted a mountain pass argument on a L 2 -sphere and constructed a special Palais-Smale sequences in searching a positive ground state as saddle point. In [22], they proved several existence and nonexistence of standing waves to (1.5) in different parameter regimes.…”

“…Due to the dipolar term, the energy functional E(u) is not invariant under rotations and this lack of symmetry prevents us from working in the radial space H 1 rad (R 3 ). In [21], the authors obtained a saddle point of E| Sc and they recovered the compactness of the corresponding Palais-Smale sequences by using the monotonicity of the mountain pass level. In [22], the authors focused on searching global minimizers of E| Sc .…”

Scattering of the energy-critical NLS with dipolar interaction

“…The nonlinear Schrödinger equations with dipolar interactions have been intensively studied in the last decade. Most works concern local existence [8], stability and instability of standing waves [1,4,17], blow-up in finite time and small data scattering [3,4,8,18]. Recently, results on scattering for "large" data were obtained [2].…”