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

Luo, Yongming; Stylianou, Athanasios

doi:10.3934/dcdsb.2020239

Cited by 4 publications

(4 citation statements)

References 59 publications

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“…Noting that F µ is even and combining with Lemma 2.6, we can suppose that u a is a nonnegative and radially symmetric decreasing function. Similar to the proof of Lemma 2.8 of [21], we get u a ∈ C 2 (R N ). Then by the strong maximum principle( [24]), we get u a > 0.…”

Section: Proof Of Theorem 12mentioning

confidence: 62%

“…Recently, normalized solutions to elliptic PDEs and systems attract much attention of researchers e.g. [15,16,20,21,25,26]. In [26], N. Soave considered the existence of normalized ground states to the following energy (Sobolev) critical Schrödinger equation…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…To prove Theorem 1.1, we follow some ideas from [21]. The main ingredient is the refined upper bound of m(a) (see Section 3), i.e.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Multiplicity and asymptotics of standing waves for the energy critical half-wave

Luo¹,

Yang²,

Xiaolong³

2021

Preprint

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In this paper, we consider the multiplicity and asymptotics of standing waves with prescribed mass R N u 2 = a 2 to the energy critical half-wavewhere N ≥ 2, a > 0, q ∈ 2, 2 + 2 N , 2 * = 2N N −1 and λ ∈ R appears as a Lagrange multiplier. We show that (0.1) admits a ground state u a and an excited state v a , which are characterised by a local minimizer and a mountain-pass critical point of the corresponding energy functional. Several asymptotic properties of {u a }, {v a } are obtained and it is worth pointing out that we get a precise description of {u a } as a → 0 + without needing any uniqueness condition on the related limit problem. The main contribution of this paper is to extend the main results in J. Bellazzini et al. [Math. Ann. 371 (2018), 707-740] from energy subcritical to energy critical case. Furthermore, these results can be extended to the general fractional nonlinear Schrödinger equation with Sobolev critical exponent, which generalize the work of H. J. Luo-Z. T. Zhang [Calc. Var. Partial Differ. Equ. 59 (2020)] from energy subcritical to energy critical case.

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Section: Proof Of Theorem 12mentioning

confidence: 62%

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Multiplicity and asymptotics of standing waves for the energy critical half-wave

Luo¹,

Yang²,

Xiaolong³

2021

Preprint

View full text Add to dashboard Cite

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“…In [7], Bellazzini and Forcella were able to utilize the ground states given in [2] and [9] to formulate a sharp scattering threshold for (1.3) without higher order term. The first results for (1.3) and (1.4) with higher order term were given by the Authors [27,28], where the cases λ 3 < 0, p = 5 and λ 3 > 0, p ∈ (4, 6] were studied. We also refer to [3,4,5,6,8,13,15,16,17,18,25,33] and the references therein for recent analytical and numerical progress on (1.3) and (1.4).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Normalized ground states for 3D dipolar Bose-Einstein condensate with attractive three-body interactions

Luo¹,

Stylianou²

2022

Preprint

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We study the existence of normalized ground states for the 3D dipolar Bose-Einstein condensate equation with attractive three-body interactions:(DBEC)When λ2 = 0 or u is radial, (DBEC) reduces to the cubic-quintic NLSwhich has been recently studied by Soave in [31]. In particular, it was shown that for any λ1 < 0 and c > 0, (CQNLS) possesses a radially symmetric ground state solution with mass c and for λ1 ≥ 0, (CQNLS) has no non-trivial solution. We show that by adding a dipole-dipole interaction to (CQNLS), the geometric nature of (CQNLS) changes dramatically and techniques as the ones from[31] cannot be used anymore to obtain similar results. More precisely, due to the axisymmetric nature of the dipole-dipole interaction potential, the energy corresponding to (DBEC) is not stable under symmetric rearrangements, hence conventional arguments based on the radial symmetry of solutions are inapplicable. We will overcome this difficulty by appealing to subtle variational and perturbative methods and prove the following:(i) If the pair (λ1, λ2) is unstable and λ1 < 0, then for any c > 0, (DBEC) has a ground state solution with mass c.(ii) If the pair (λ1, λ2) is unstable and λ1 ≥ 0, then there exists some c * = c * (λ1, λ2) ≥ 0 such that for all c > c * , (DBEC) has a ground state solution with mass c. Moreover, any non-trivial solution of (DBEC) in this case must be non-radial.(iii) If the pair (λ1, λ2) is stable, then (DBEC) has no non-trivial solutions.

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Stabilization in dipolar Gross–Pitaevskii theory by mass-subcritical perturbation

Luo

Yang

2022

J. Fixed Point Theory Appl.

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

Cited by 4 publications

References 59 publications

Multiplicity and asymptotics of standing waves for the energy critical half-wave

Multiplicity and asymptotics of standing waves for the energy critical half-wave

Normalized ground states for 3D dipolar Bose-Einstein condensate with attractive three-body interactions

Stabilization in dipolar Gross–Pitaevskii theory by mass-subcritical perturbation

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