Symmetry Results for Semilinear Elliptic Systems in the Whole Space

Busca, Jérôme; Sirakov, Boyan

doi:10.1006/jdeq.1999.3701

Cited by 177 publications

(159 citation statements)

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“…For the definition of cooperative systems and applications of the method of moving plane to cooperative systems, we refer to Troy [22] (for bounded domains) and Busca-Sirakov [5] (for the whole R N ). See also Lin-Zhang [16] for the Liouville system which is cooperative.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic behavior of SU(3) Toda system in a bounded domain

2011

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Section: Introductionmentioning

confidence: 99%

Asymptotic behavior of SU(3) Toda system in a bounded domain

2011

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“…We mention that in case p(N −2), q(N −2) < (N +2) and p, q > 1, then it follows from [17, Theorem 2.1] that solutions decay at infinity; as a consequence, according to [6,Theorem 2], in this case both u and v are radially symmetric and radially decreasing with respect to some point. We do not know whether this conclusion holds under the mere assumptions of Theorem 1.9.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Ground state and non-ground state solutions of some strongly coupled elliptic systems

Bonheure

Santos²,

Ramos

2011

Trans. Amer. Math. Soc.

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Abstract. We study an elliptic system of the form Lu = |v| p−1 v and Lv = |u| q−1 u in Ω with homogeneous Dirichlet boundary condition, where Lu := −Δu in the case of a bounded domain and Lu := −Δu + u in the cases of an exterior domain or the whole space R N . We analyze the existence, uniqueness, sign and radial symmetry of ground state solutions and also look for sign changing solutions of the system. More general non-linearities are also considered.

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“…This follows similarly as in [16, Theorem 2.1] and therefore we omit the proof. We just mention that the conclusion relies on the fact that every positive solutions u, v of the system (3.4) lying in H 1 (R N ) are radially symmetric with respect to some point z 0 ∈ R N (see [4,Theorem 2]). The proof also uses an argument based on the information on the Morse index of the solutions (u j , v j ), similar to the one in Lemma 3.3 below.…”

Section: Lemma 31 We Have That ρ J → ∞ As J → ∞mentioning

confidence: 99%

Locating the peaks of the least energy solutions to an elliptic system with Dirichlet boundary conditions

Pistoia

Ramos

2008

Nonlinear differ. equ. appl.

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Abstract. We consider a system of the form −ε 2 ∆u + u = g(v), −ε 2 ∆v + v = f (u) in Ω with Dirichlet boundary condition on ∂Ω, where Ω is a smooth bounded domain in R N , N 3 and f, g are power-type nonlinearities having superlinear and subcritical growth at infinity. We prove that the least energy solutions to such a system concentrate, as ε goes to zero, at a point of Ω which maximizes the distance to the boundary of Ω.2000 Mathematics Subject Classification: 35J50, 35J55, 38E05.

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Symmetry Results for Semilinear Elliptic Systems in the Whole Space

Cited by 177 publications

References 16 publications

Asymptotic behavior of SU(3) Toda system in a bounded domain

Asymptotic behavior of SU(3) Toda system in a bounded domain

Ground state and non-ground state solutions of some strongly coupled elliptic systems

Locating the peaks of the least energy solutions to an elliptic system with Dirichlet boundary conditions

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