# Monotonicity and symmetry of nonnegative solutions to $ -\Delta u=f(u) $ in half-planes and strips

## TL;DR

This paper establishes symmetry and monotonicity of nonnegative solutions to $- riangle u=f(u)$ in half-planes and strips using a novel rotating and sliding line technique, accommodating very general nonlinearities including non-Lipschitz cases.

## Contribution

It introduces a unified method for proving symmetry and monotonicity for a broad class of nonlinearities in unbounded domains.

## Key findings

- Proves symmetry and monotonicity under general conditions on $f$.
- Handles nonlinearities that are not locally Lipschitz.
- Provides examples demonstrating the sharpness of assumptions.

## Abstract

We consider nonnegative solutions to $-\Delta u=f(u)$ in half-planes and strips, under zero Dirichlet boundary condition. Exploiting a rotating$\&$sliding line technique, we prove symmetry and monotonicity properties of the solutions, under very general assumptions on the nonlinearity $f$. In fact we provide a unified approach that works in all the cases $f(0)<0$, $f(0)= 0$ or $f(0)> 0$. Furthermore we make the effort to deal with nonlinearities $f$ that may be not locally-Lipschitz continuous. We also provide explicite examples showing the sharpness of our assumptions on the nonlinear function $f$.