Qualitative properties and classification of nonnegative solutions to in unbounded domains when
Alberto Farina
Université de Picardie Jules Verne, Amiens, FranceBerardino Sciunzi
Università della Calabria, Arcavacata di Rende, Italy
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Abstract
We consider nonnegative solutions to in unbounded Euclidean domains under zero Dirichlet boundary conditions, where is merely locally Lipschitz continuous and satisfies . In the half-plane, and without any other assumption on , we prove that 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.
Cite this article
Alberto Farina, Berardino Sciunzi, Qualitative properties and classification of nonnegative solutions to in unbounded domains when . Rev. Mat. Iberoam. 32 (2016), no. 4, pp. 1311–1330
DOI 10.4171/RMI/918