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Qualitative properties and classification of nonnegative solutions to -Delta u = f (u) in unbounded domains when f (0) < 0

Abstract : We consider nonnegative solutions to -Delta u = f (u) in unbounded Euclidean domains under zero Dirichlet boundary conditions, where f is merely locally Lipschitz continuous and satisfies f (0) < 0. In the half-plane, and without any other assumption on u, we prove that u is either one-dimensional and periodic or positive and strictly monotone increasing in the direction orthogonal to the boundary. Analogous results are obtained if the domain is a strip. As a consequence of our main results, we answer affirmatively to a conjecture and to an open question posed by Berestycki, Caffarelli and Nirenberg. We also obtain some symmetry and monotonicity results in the higher-dimensional case.
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https://hal-u-picardie.archives-ouvertes.fr/hal-03621424
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Soumis le : lundi 28 mars 2022 - 10:57:13
Dernière modification le : mardi 29 mars 2022 - 03:58:29

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Alberto Farina, Berardino Sciunzi. Qualitative properties and classification of nonnegative solutions to -Delta u = f (u) in unbounded domains when f (0) < 0. REVISTA MATEMATICA IBEROAMERICANA, 2016, 32 (4), pp.1311-1330. ⟨10.4171/RMI/918⟩. ⟨hal-03621424⟩

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