Existence of least energy positive solutions to critical Schrödinger systems in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e22" altimg="si12.svg"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math>

You, Song; Zou, Wenming

doi:10.1016/j.aml.2021.107900

Cited by 3 publications

(2 citation statements)

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“…All the papers that deal with system (1.1) with d ≥ 2 in the critical case (2p = 2 * ) mainly focus on the higher dimensional case (N ≥ 4). To the best of our knowledge, there are only three papers [20,38,40] studying system (1.1) for the critical case on a smooth bounded domain with N = 3 and d = 2 in the literature. In [20,38] the authors proved that there exists β > 0 such that system (1.1) has a least energy positive solution if β 12 > β.…”

Section: Introductionmentioning

confidence: 99%

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Least energy positive soultions for $d$-coupled Schrödinger systems with critical exponent in dimension three

Liu¹,

Song²,

Zou³

2022

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In the present paper, we consider the coupled Schrödinger systems with critical exponent:Here, Ω ⊂ R 3 is a smooth bounded domain, d ≥ 2, β ii > 0 for every i, and, where λ 1 (Ω) is the first eigenvalue of −∆ with Dirichlet boundary conditions and λ * (Ω) ∈ (0, λ 1 (Ω)). We acquire the existence of least energy positive solutions to this system for weakly cooperative case (β ij > 0 small) and for purely competitive case (β ij ≤ 0) by variational arguments. The proof is performed by mathematical induction on the number of equations, and requires more refined energy estimates for this system. Besides, we present a new nonexistence result, revealing some different phenomena comparing with the higher-dimensional case N ≥ 5. It seems that this is the first paper to give a rather complete picture for the existence of least energy positive solutions to critical Schrödinger system in dimension three.

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Section: Introductionmentioning

confidence: 99%

“…In [20,38] the authors proved that there exists β > 0 such that system (1.1) has a least energy positive solution if β 12 > β. Recently, You and Zou [40] proved that system (1.1) has a least energy positive solution for β 12 > 0 small. Those papers mentioned above only deal with the purely cooperative case (β 12 > 0).…”

Section: Introductionmentioning

confidence: 99%