On symmetry of nonnegative solutions of elliptic equations
نویسندگان
چکیده
منابع مشابه
On symmetry of nonnegative solutions of elliptic equations
We consider the Dirichlet problem for a class of fully nonlinear elliptic equations on a bounded domain Ω. We assume that Ω is symmetric about a hyperplane H and convex in the direction perpendicular to H. By a well-known result of Gidas, Ni and Nirenberg and its generalizations, all positive solutions are reflectionally symmetric about H and decreasing away from the hyperplane in the direction...
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We consider the Dirichlet problem for the semilinear equation ∆u + f(u) = 0 on a bounded domain Ω ⊂ RN . We assume that Ω is convex in a direction e and symmetric about the hyperplane H = {x ∈ RN : x · e = 0}. It is known that if N ≥ 2 and Ω is of class C2, then any nonzero nonnegative solution is necessarily strictly positive and, consequently, it is reflectionally symmetric about H and decrea...
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We consider the Dirichlet problem for semilinear elliptic equations on a smooth bounded domain Ω. We assume that Ω is symmetric about a hyperplane H and convex in the direction orthogonal to H. Employing Serrin’s result on an overdetermined problem, we show that any nonzero nonnegative solution is necessarily strictly positive. One can thus apply a well-known result of Gidas, Ni and Nirenberg t...
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In a recent interesting paper, GIDAS, NI, and NIRENBERG [2] proved that positive solutions of the Dirichlet problem for second-order semi-linear elliptic equations on balls must themselves be spherically symmetric functions. Here we consider the bifurcation problem for such solutions. Specifically, we investigate the ways in which the symmetric solution can bifurcate into a nonsymmetric solutio...
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ژورنال
عنوان ژورنال: Annales de l'Institut Henri Poincaré C, Analyse non linéaire
سال: 2012
ISSN: 0294-1449
DOI: 10.1016/j.anihpc.2011.03.001