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

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, stands for the fractional laplacian, (Nge 2), ({mathbb {R}}^N_+={x=(x',x_N)in {mathbb {R}}^N{:} x_N>0}) is the half-space and (fin 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).