  We establish existence of positive non-decreasing radial solutions for a
nonlocal nonlinear Neumann problem both in the ball and in the annulus. The
nonlinearity that we consider is rather general, allowing for supercritical
growth (in the sense of Sobolev embedding). The consequent lack of compactness
can be overcome, by working in the cone of non-negative and non-decreasing
radial functions. Within this cone, we establish some a priori estimates which
allow, via a truncation argument, to use variational methods for proving
existence of solutions. As a side result, we prove a strong maximum principle
for nonlocal Neumann problems, which is of independent interest.
