Monotonicity in half-spaces of positive solutions to $-\Delta_p u=f(u)$ in the case $p>2$

TL;DR

This paper proves that positive solutions to a class of p-Laplace equations in half-spaces are strictly monotone increasing in the direction orthogonal to the boundary for p>2, leading to Liouville theorems and regularity results.

Contribution

It establishes monotonicity results for positive solutions of $- abla_p u=f(u)$ in half-spaces specifically for p>2, extending previous knowledge from the case 1<p≤2.

Findings

Positive solutions are strictly monotone increasing in the orthogonal direction for p>2.

Nonnegative solutions are $C^{2,eta}$ smooth.

Derived Liouville type theorems for the equation.

Abstract

We consider weak distributional solutions to the equation $-\Delta_pu=f(u)$ in half-spaces under zero Dirichlet boundary condition. We assume that the nonlinearity is positive and superlinear at zero. For $p>2$ (the case $1<p\leq2$ is already known) we prove that any positive solution is strictly monotone increasing in the direction orthogonal to the boundary of the half-space. As a consequence we deduce some Liouville type theorems for the Lane-Emden type equation. Furthermore any nonnegative solution turns out to be $C^{2,\alpha}$ smooth.