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Further qualitative properties for elliptic equations in unbounded domains

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Further qualitative properties for elliptic equations in unbounded domains ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze HENRIBERESTYCKI LUISCAFFARELLI LOUISNIRENBERG Further qualitative properties for elliptic equations in unbounded domains Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e série, tome 25, no 1-2 (1997), p. 69-94. <http://www.numdam.org/item?id=ASNSP_1997_4_25_1-2_69_0> © Scuola Normale Superiore, Pisa, 1997, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 69 Further Qualitative Properties for Elliptic Equations in Unbounded Domains HENRI BERESTYCKI - LUIS CAFFARELLI - LOUIS NIRENBERG Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) Vol. XXV (1997), pp. 69-94 Dedicated to the Memory of Ennio De Giorgi 1. - Introduction and main results This article is one in a series by the authors to study some qualitative properties of positive solutions of elliptic second order boundary value problems of the type in various kinds of unbounded domains Q of R n. Typically, we are interested in features like monotonicity in some directions and symmetry. In some cases, the positive solutions we consider are supposed to be bounded while in other cases boundedness is not assumed. The function f appearing in (1.1) will always be assumed to be (globally) Lipschitz continuous: JR+ ~ R. The present paper is devoted to the investigation of three main configu- rations. We consider a half space S2 = fx = (xi, ... ,

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