Monotonicity of solutions for some nonlocal elliptic problems in half-spaces

TL;DR

This paper proves that bounded, nonnegative solutions to a fractional elliptic problem in a half-space are strictly positive and monotone increasing in the normal direction, extending known results from the classical case to fractional operators.

Contribution

It establishes the monotonicity and positivity of solutions for fractional elliptic problems in half-spaces without restrictions on the nonlinearity.

Findings

Solutions are positive in the half-space.

Solutions are strictly increasing in the normal direction.

Results contrast with classical case where non-positivity solutions exist.

Abstract

In this paper we consider classical solutions $u$ of the semilinear fractional problem $(-\Delta)^s u = f(u)$ in $\mathbb{R}^N_+$ with $u=0$ in $\mathbb{R}^N \setminus \mathbb{R}^N_+$, where $(-\Delta)^s$, $0<s<1$, stands for the fractional laplacian, $N\ge 2$, $\mathbb{R}^N_+=\{x=(x',x_N)\in \mathbb{R}^N:\ x_N>0\}$ is the half-space and $f\in C^1$ is a given function. With no additional restriction on the function $f$, we show that bounded, nonnegative, nontrivial classical solutions are indeed positive in $\mathbb{R}^N_+$ and verify $$ \frac{\partial u}{\partial x_N}>0 \quad \hbox{in } \mathbb{R}^N_+. $$ This is in contrast with previously known results for the local case $s=1$, where nonnegative solutions which are not positive do exist and the monotonicity property above is not known to hold in general even for positive solutions when $f(0)<0$.