On Coron's problem for weakly coupled elliptic systems

Pistoia, Angela; Soave, Nicola

doi:10.1112/plms.12073

Cited by 24 publications

(21 citation statements)

References 29 publications

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“…Theorem 1.2 extends some earlier results obtained in [5,6] for a system of two equations; see also [9]. Existence and multiplicity results for the purely critical system in a bounded domain may be found in [5,13,14]. Supercritical systems were recently considered in [4].…”

Section: Introductionsupporting

confidence: 75%

A simple variational approach to weakly coupled competitive elliptic systems

Clapp

Szulkin

2019

Nonlinear Differ. Equ. Appl.

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The main purpose of this paper is to exhibit a simple variational setting for finding fully nontrivial solutions to the weakly coupled elliptic system (1.1). We show that such solutions correspond to critical points of a C 1 -functional Ψ : U → R defined in an open subset U of the product T := S1 × · · · × SM of unit spheres Si in an appropriate Sobolev space. We use our abstract setting to extend and complement some known results for the system (1.1).Keywords: Weakly coupled elliptic system, simple variational setting, subcritical system in exterior domain, entire solutions to critical system, Brezis-Nirenberg problem.

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Section: Introductionsupporting

confidence: 75%

A simple variational approach to weakly coupled competitive elliptic systems

Clapp

Szulkin

2019

Nonlinear Differ. Equ. Appl.

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“…Proof. We reason similarly to [20,Lemma 4.3], to which we refer for more details. First of all, using (1.6), we have that 1 2Ω…”

Section: Expansion Of the Reduced Energymentioning

confidence: 99%

“…Then, for A =´R N U 3 1,0 = α4 γ4 and R defined by The following concerns the asymptotic study of L q norms of the bubble. For the proof, see for instance [20,Lemma A.3] if q = 4.…”

Section: Appendix a Asymptotic Estimatesmentioning

confidence: 99%

A fountain of positive bubbles on a Coron's problem for a competitive weakly coupled gradient system

Pistoia

Soave

Tavares

2020

Journal de Mathématiques Pures et Appliquées

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We consider the following critical elliptic system:in a domain Ωε ⊂ R 4 with a small shrinking hole Bε(ξ 0 ). For µ i > 0, β < 0, and ε > 0 small, we prove the existence of a non-synchronized solution which looks like a fountain of positive bubbles, i.e. each component u i exhibits a towering blow-up around ξ 0 as ε → 0. The proof is based on the Ljapunov-Schmidt reduction method, and the velocity of concentration of each layer within a given tower is chosen in such a way that the interaction between bubbles of different components balance the interaction of the first bubble of each component with the boundary of the domain, and in addition is dominant when compared with the interaction of two consecutive bubbles of the same component.

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“…In [9,13,21], the authors investigated the existence and multiplicity of fully nontrivial solutions to (1) with λ i = 0 for the critical case in R N . Recently [23,24] are concerned with existence and concentration results for a Coron-type problem in a bounded domain with one or multiple small holes in the case λ i = 0. In [5], the first author with D. Cassani and J. Zhang studied the existence of least energy positive solutions in the critical exponential case when N = 2.…”

Section: Introductionmentioning

confidence: 99%

Existence of least energy positive solutions to Schrödinger systems with mixed competition and cooperation terms: the critical case

Tavares

You

2020

Calc. Var.

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In this paper we investigate the existence of solutions to the following Schrödinger system in the critical caseHere, Ω ⊂ R 4 is a smooth bounded domain, d ≥ 2, −λ1(Ω) < λi < 0 and βii > 0 for every i, βij = βji for i = j, where λ1(Ω) is the first eigenvalue of −∆ with Dirichlet boundary conditions. Under the assumption that the components are divided into m groups, and that βij ≥ 0 (cooperation) whenever components i and j belong to the same group, while βij < 0 or βij is positive and small (competition or weak cooperation) for components i and j belonging to different groups, we establish the existence of nonnegative solutions with m nontrivial components, as well as classification results. Moreover, under additional assumptions on βij , we establish existence of least energy positive solutions in the case of mixed cooperation and competition. The proof is done by induction on the number of groups, and requires new estimates comparing energy levels of the system with those of appropriate sub-systems. In the case Ω = R 4 and λ1 = . . . = λm = 0, we present new nonexistence results. This paper can be seen as the counterpart of [Soave-Tavares, J. Differential Equations 261 (2016), 505-537] in the critical case, while extending and improving some results from [Chen-Zou, Arch.

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On Coron's problem for weakly coupled elliptic systems

Cited by 24 publications

References 29 publications

A simple variational approach to weakly coupled competitive elliptic systems

A simple variational approach to weakly coupled competitive elliptic systems

A fountain of positive bubbles on a Coron's problem for a competitive weakly coupled gradient system

Existence of least energy positive solutions to Schrödinger systems with mixed competition and cooperation terms: the critical case

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