  We prove the existence of positive solutions for the supercritical nonlinear
fractional Schr\"odinger equation $(-\Delta)^s u+V(x)u-u^p=0$ in $\mathbb R^n$,
with $u(x)\to 0$ as $|x|\to +\infty$, where $p>\frac{n+2s}{n-2s}$ for $s\in
(0,1), \ n>2s$. We show that if $V(x)=o(|x|^{-2s})$ as $|x|\to +\infty$, then
for $p>\frac{n+2s-1}{n-2s-1}$, this problem admits a continuum of solutions.
More generally, for $p>\frac{n+2s}{n-2s}$, conditions for solvability are also
provided. This result is the extension of the work by Davila, Del Pino, Musso
and Wei to the fractional case. Our main contributions are: the existence of a
smooth, radially symmetric, entire solution of $(-\Delta)^s w=w^p$ in $\mathbb
R^n$, and the analysis of its properties. The difficulty here is the lack of
phase-plane analysis for a nonlocal ODE; instead we use conformal geometry
methods together with Schaaf's argument as in the paper by Ao, Chan, DelaTorre,
Fontelos, Gonz\'alez and Wei on the singular fractional Yamabe problem.
