Revista Matemática Iberoamericana


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Volume 32, Issue 4, 2016, pp. 1311–1330
DOI: 10.4171/RMI/918

Published online: 2016-12-16

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

Alberto Farina[1] and Berardino Sciunzi[2]

(1) Université de Picardie Jules Verne, Amiens, France
(2) Università della Calabria, Arcavacata di Rende, Italy

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.

Keywords: Semilinear elliptic equations, qualitative properties of the solutions, moving plane method

Farina Alberto, Sciunzi Berardino: Qualitative properties and classification of nonnegative solutions to $-\Delta u=f(u)$ in unbounded domains when $f(0) < 0$. Rev. Mat. Iberoamericana 32 (2016), 1311-1330. doi: 10.4171/RMI/918