  We provide fine asymptotics of solutions of fractional elliptic equations at
boundary points where the domain is locally conical; that is, corner type
singularities appear. Our method relies on a suitable smoothing of the corner
singularity and an approximation scheme, which allow us to provide a Pohozaev
type inequality. Then, the asymptotics of solutions at the conical point follow
by an Almgren type monotonicity formula, blow-up analysis and Fourier
decomposition on eigenspaces of a spherical eigenvalue problem. A strong unique
continuation principle follows as a corollary.
