Ground state in the energy super-critical Gross–Pitaevskii equation with a harmonic potential

Bizoń, Piotr; Ficek, Filip; Pelinovsky, Dmitry E.; Sobieszek, Szymon

doi:10.1016/j.na.2021.112358

Cited by 9 publications

(23 citation statements)

References 19 publications

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“…On the other hand, energy-critical p = 2d d−2 and energy-supercritical p > 2d d−2 cases were less investigated in the literature (cf. [33][34][35], [4,14,29], and [10]).…”

Section: Introductionmentioning

confidence: 99%

“…By the method of Lagrange's multipliers and the scaling transformation, u = (I ω ) d−2 4 v is a nontrivial solution of the stationary equation (1.1) if v is a minimizer of the variational problem (1.4). Based on the above observations, we can introduce the following definition.…”

mentioning

confidence: 99%

“…In the energy-supercritical case p > 2d d−2 , we will fix p = 4 to simplify the computations similarly to what was adopted in [4,29]. The energy-supercritical case for p = 4 corresponds to d ≥ 5 and the stationary equation (1.1) is reduced to…”

mentioning

confidence: 99%

“…It has been proved in [35] (see also [4] for a different proof), that there exists a singular radial solution u ∞ of the stationary equation (1.12) for some ω…”

mentioning

confidence: 99%

“…Let v ω be the minimizer of the variational problem (1.4) for ω ∈ (ω * , d). Then, 4 v ω is the ground state solution of the stationary equation (1.1). Since we are interested in ω → ω * with ω * < d, it is standard to show that {u ω } is bounded in X.…”

mentioning

confidence: 99%

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Positive solutions of the Gross-Pitaevskii equation for energy critical and supercritical nonlinearities

Pelinovsky¹,

Wei²,

Wu³

2022

Preprint

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We consider positive and spatially decaying solutions to the following Gross-Pitaevskii equation with a harmonic potential:where d ≥ 3, p > 2 and ω > 0. For p = 2d d−2 (energy-critical case) there exists a ground state u ω if and only if ω ∈ (ω * , d), where ω * = 1 for d = 3 and ω * = 0 for d ≥ 4. We give a precise description on asymptotic behaviors of u ω as ω → ω * up to the leading order term for different values of d ≥ 3. When p > 2d d−2 (energy-supercritical case) there exists a singular solution u ∞ for some ω ∈ (0, d). We compute the Morse index of u ∞ in the class of radial functions and show that the Morse index of u ∞ is infinite in the oscillatory case, is equal to 1 or 2 in the monotone case for p not large enough and is equal to 1 in the monotone case for p sufficiently large.

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“…On the other hand, energy-critical p = 2d d−2 and energy-supercritical p > 2d d−2 cases were less investigated in the literature (cf. [33][34][35], [4,14,29], and [10]).…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…It has been proved in [35] (see also [4] for a different proof), that there exists a singular radial solution u ∞ of the stationary equation (1.12) for some ω…”

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Positive solutions of the Gross-Pitaevskii equation for energy critical and supercritical nonlinearities

Pelinovsky¹,

Wei²,

Wu³

2022

Preprint

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Morse index for the ground state in the energy supercritical Gross–Pitaevskii equation

Pelinovsky

Sobieszek

2022

Journal of Differential Equations

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Positive solutions of the Gross–Pitaevskii equation for energy critical and supercritical nonlinearities

Pelinovsky

Wei

2023

Nonlinearity

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We consider positive and spatially decaying solutions to the following Gross–Pitaevskii equation with a harmonic potential: − Δ u + | x | 2 u = ω u + | u | p − 2 u in R d , where d ⩾ 3 , p > 2 and ω > 0. For p = 2 d d − 2 (energy-critical case) there exists a ground state u ω if and only if ω ∈ ( ω ∗ , d ) , where ω ∗ = 1 for d = 3 and ω ∗ = 0 for d ⩾ 4 . We give a precise description on asymptotic behaviours of u ω as ω → ω ∗ up to the leading order term for different values of d ⩾ 3 . When p > 2 d d − 2 (energy-supercritical case) there exists a singular solution u ∞ for some ω ∈ ( 0 , d ) . We compute the Morse index of u ∞ in the class of radial functions and show that the Morse index of u ∞ is infinite in the oscillatory case, is equal to 1 or 2 in the monotone case for p not large enough and is equal to 1 in the monotone case for p sufficiently large.

show abstract

Ground state in the energy super-critical Gross–Pitaevskii equation with a harmonic potential

Cited by 9 publications

References 19 publications

Positive solutions of the Gross-Pitaevskii equation for energy critical and supercritical nonlinearities

Positive solutions of the Gross-Pitaevskii equation for energy critical and supercritical nonlinearities

Morse index for the ground state in the energy supercritical Gross–Pitaevskii equation

Positive solutions of the Gross–Pitaevskii equation for energy critical and supercritical nonlinearities

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