2008
DOI: 10.1016/j.anihpc.2006.11.006
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Solitary waves for some nonlinear Schrödinger systems

Abstract: In this paper we study the existence of radially symmetric positive solutions in H 1 rad (R N) × H 1 rad (R N) of the elliptic system: − u + u − αu 2 + βv 2 u = 0, − v + ω 2 v − βu 2 + γ v 2 v = 0, N = 1, 2, 3, where α and γ are positive constants (β will be allowed to be negative). This system has trivial solutions of the form (φ, 0) and (0, ψ) where φ and ψ are nontrivial solutions of scalar equations. The existence of nontrivial solutions for some values of the parameters α, β, γ , ω has been studied recent… Show more

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Cited by 73 publications
(54 citation statements)
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“…d) Extensions of Sirakov's results were also obtained, in particular, in the one dimensional case in [7].…”
mentioning
confidence: 77%
“…d) Extensions of Sirakov's results were also obtained, in particular, in the one dimensional case in [7].…”
mentioning
confidence: 77%
“…Moreover, the authors gave sufficient conditions for ground states to be positive in both components which basically require the coupling parameter τ to be positive and sufficiently large. Also, Ambrosetti and Colorado [5] and de Figueiredo and Lopes [16] have proved the additional sufficient conditions for the existence of positive ground-state solutions in the special case p = r = 2q = 4. Furthermore, for p = r = 2q = 4 and small positive values of τ, Lin and Wei [25] and Sirakov [36] proved the existence of positive solutions which have minimal energy among all fully nontrivial solutions.…”
Section: Introductionmentioning
confidence: 99%
“…In particular, the case where V 1 (x), V 2 (x) are positive and independent of x is well studied and it is shown in [1,3,[8][9][10]13] that there exist positive constants β 2 ≥ β 1 > 0 such that for β ∈ [0, β 1 ) ∪ (β 2 , ∞), (1)-(4) has a nontrivial positive solution. And it has been extended to x-dependent situations in [6,14].…”
Section: Introductionmentioning
confidence: 99%
“…The system (5) arises in many physical problems, especially in the Hartree-Fock theory and nonlinear optics. We refer to [1,3,6,[8][9][10]13,14] and references therein.…”
Section: Introductionmentioning
confidence: 99%
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