Existence and non-existence results for a fractional Lane-Emden equation with nonlocal Neumann conditions

We consider the fractional Lane-Emden equation with a nonlocal Neumann condition in a half-space. We establish the existence of non-constant solutions for the critical problem in any dimension. On the other hand, we show that, in dimension $n=1$, the subcritical problem admits only the trivial solution. This result follows from a new Pohozaev-type identity, which we obtain by using suitable decay estimates for the solutions.