Existence and non-existence results for a semilinear fractional Neumann problem

We establish a priori $L^{\mathrm{\infty }}$-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 $0<s\le 1\mathrm{/}2$ the analysis started in [7].