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 R-+(N) with u = 0 in R-N\R-+(N), where (-Delta)(s), 0 < s < 1, stands for the fractional laplacian, N >= 2, R-+(N) = {x = (x ', xN) is an element of R-N : x(N) > 0} is the half-space and f is an element of 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 R-+(N) and verify partial derivative u/partial derivative x(N) > 0 in 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.