In this paper we use the moving plane method to get the radial symmetry about a point x 0 ∈ R N of the positive ground state solutions of the equationWe assume f to be locally Lipschitz continuous in (0, +∞) and nonincreasing near zero but we do not require any hypothesis on the critical set of the solution. To apply the moving plane method we first prove a weak comparison theorem for solutions of differential inequalities in unbounded domains.
L'accès aux archives de la revue « Annales de l'I. H. P., section C » (http://www.elsevier.com/locate/anihpc) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation 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/ Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principleL ucio DAMASCELLI (1), Massimo GROSSI (2) and Filomena PACELLA (3)
In this paper, we study the symmetry properties of the solutions of the semilinear elliptic problemwhere O is a bounded symmetric domain in R N , N 52, and f : O Â R ! R is a continuous function of class C 1 in the second variable, g is continuous and f and g are somehow symmetric in x. Our main result is to show that all solutions of the above problem of index one are axially symmetric when O is an annulus or a ball, g 0 and f is strictly convex in the second variable. To do this, we prove that the nonnegativity of the first eigenvalue of the linearized operator in the caps determined by the symmetry of O is a sufficient condition for the symmetry of the solution, when f is a convex function. # 2002 Elsevier Science (USA)
We consider a partially overdetermined problem in a sector-like domain Ω in a cone Σ in R N , N ≥ 2, and prove a rigidity result of Serrin type by showing that the existence of a solution implies that Ω is a spherical sector, under a convexity assumption on the cone. We also consider the related question of characterizing constant mean curvature compact surfaces Γ with boundary which satisfy a 'gluing' condition with respect to the cone Σ. We prove that if either the cone is convex or the surface is a radial graph then Γ must be a spherical cap. Finally we show that, under the condition that the relative boundary of the domain or the surface intersects orthogonally the cone, no other assumptions are needed.2010 Mathematics Subject Classification. 35N25, 35B06, 53A10, 53A05.
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