  We establish a priori $L^\infty$-estimates for non-negative solutions of a
semilinear nonlocal Neumann problem. As a consequence of these estimates, we
get non-existence of non-constant solutions under suitable assumptions on the
diffusion coefficient and on the nonlinearity. Moreover, we prove an existence
result for radial, radially non-decreasing solutions in the case of a possible
supercritical nonlinearity, extending to the case $0<s\le 1/2$ the analysis
started in [7].
