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

TL;DR

This paper investigates the qualitative behavior of nonnegative solutions to the nonlinear PDE $- riangle u=f(u)$ in unbounded domains, establishing conditions for symmetry, monotonicity, and periodicity, and confirming conjectures in the field.

Contribution

It proves that solutions in half-planes and strips are either one-dimensional and periodic or monotone, confirming a conjecture and answering an open question by Berestycki, Caffarelli, and Nirenberg.

Findings

Solutions are either one-dimensional and periodic or monotone increasing.

Confirmed a conjecture and answered an open question in the literature.

Extended symmetry and monotonicity results to higher-dimensional cases.

Abstract

We consider nonnegative solutions to $-\Delta u=f(u)$ in unbounded euclidean domains, 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.