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

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

$$\begin{aligned} \frac{\partial u}{\partial x_N}>0 \quad \hbox {in } {\mathbb {R}}^N_+. \end{aligned}$$

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 $f(0)<0$.