Normalized solutions for Schrödinger equations with critical Sobolev exponent and mixed nonlinearities

Wei, Juncheng; Wu, Yuanze

doi:10.1016/j.jfa.2022.109574

Cited by 92 publications

(44 citation statements)

References 69 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The monotone property was first pointed out by Brezís and Nirenberg in [6,Remark 1.5]. The detailed proofs were recently given in [9, Lemma 2.1] and [41,Lemma 3.3]. Hence, we have 0 < I ω < S = I ω * for all ω ∈ (ω * , d).…”

mentioning

confidence: 79%

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

Pelinovsky¹,

Wei²,

Wu³

2022

Preprint

View full text Add to dashboard Cite

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.

show abstract

mentioning

confidence: 79%

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

Pelinovsky¹,

Wei²,

Wu³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…In recent years, much attention has been paid to the study of normalized solutions of nonlinear Schrödinger equations, see for example [9,10,11,12] and the references therein. For some related studies on the mixeddispersion fourth-order Schrödinger equations, we refer the reader to [13,14,15].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Normalized ground states of nonlinear biharmonic Schrödinger equations with Sobolev critical growth and combined nonlinearities

Chang

2023

Applied Mathematics Letters

View full text Add to dashboard Cite

“…It is worth pointing out that, Jeanjean and Le [21] solved this open question for N ≥ 4. Recently, Wei and Wu [39] proved the existence of normalized solutions of mountainpass type of (1.5) for N ≥ 3 and 2 < q < 2 + 4 N . Moreover, in [39], they also got the precisely asymptotic behaviors of ground states and mountain-pass solutions as µ → 0 or µ goes to its upper bound.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Prescribed mass solutions to Schrödinger systems with critical Sobolev exponent

Luo

Xiaolong

Zou

2022

Preprint

View full text Add to dashboard Cite

Note: Please see pdf for full abstract with equations. We consider the existence and asymptotic behavior on mass of positive solutions to the following system −Δu + λ1u = μ1u3 + α1|u|p−2u + βv2u in R4, −Δv + λ2v = μ2v3 + α2|v|p−2v + βu2v in R4, under the normalized mass constraints ∫R4u2 = a21 and ∫R4v2 = a22, where a1, a2 are prescribed, μ1, μ2, β > 0, α1, α2 ∈ R, p ∈ (2, 4) and λ1, λ2 ∈ R appear as Lagrange multipliers. Firstly, we establish a non-existence result for the repulsive interaction case, i.e., αi < 0(i = 1, 2). Then turning to the case of αi > 0(i = 1, 2), if 2 < p < 3, we show that the problem admits a ground state and an excited state. Moreover, we give a precise asymptotic behavior of these two solutions as (a1, a2) → (0, 0) and a1 ~ a2. This seems to be the first contribution regarding the multiplicity as well as synchronized mass collapse behavior of normalized solutions to Schrödinger systems with Sobolev critical exponent. When 3 ≤ p < 4, we prove an existence as well as a non-existence (p = 3) results of ground states. Furthermore, precise asymptotic behaviors of the ground states are obtained when the masses of whose two components vanish and cluster to a upper bound (or infinity), respectively. 2010 Mathematics Subject Classification : 35J50, 35B33, 35B09, 35B40.

show abstract

Normalized solutions for Schrödinger equations with critical Sobolev exponent and mixed nonlinearities

Cited by 92 publications

References 69 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

Normalized ground states of nonlinear biharmonic Schrödinger equations with Sobolev critical growth and combined nonlinearities

Prescribed mass solutions to Schrödinger systems with critical Sobolev exponent

Contact Info

Product

Resources

About