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 (- Δ) su= f(u) in R+N with u= 0 in RNR+N, where (- Δ) s, 0 < s< 1 , stands for the fractional laplacian, N≥ 2 , R+N={x=(x′,xN)∈RN:xN>0} is the half-space and f∈ C1 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 (Formula presented.). 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.