Monotonicity of positive solutions to quasilinear elliptic equations in half-spaces with a changing-sign nonlinearity

TL;DR

This paper proves that positive solutions to certain quasilinear elliptic equations in half-spaces are monotonic, using advanced comparison and maximum principles along with an adapted moving plane method for unbounded domains.

Contribution

It establishes monotonicity of solutions for a broad class of nonlinearities in quasilinear elliptic equations, extending previous results to changing-sign nonlinearities in unbounded domains.

Findings

Positive solutions are monotonic in half-spaces.

The method adapts the moving plane technique to unbounded domains.

Results apply to a wide class of nonlinearities with sign changes.

Abstract

In this paper we prove the monotonicity of positive solutions to $ -\Delta_p u = f(u) $ in half-spaces under zero Dirichlet boundary conditions, for $(2N+2)/(N+2) < p < 2$ and for a general class of regular changing-sign nonlinearities $f$. The techniques used in the proof of the main result are based on a fine use of comparison and maximum principles and on an adaptation of the celebrated moving plane method to quasilinear elliptic equations in unbounded domains.