  We consider a nonlinear pseudo-differential equation driven by the fractional
$p$-Laplacian $(-\Delta)^s_p$ with $s\in(0,1)$ and $p\ge 2$ (degenerate case),
under Dirichlet type conditions in a smooth domain $\Omega$. We prove that
local minimizers of the associated energy functional in the fractional Sobolev
space $W^{s,p}_0(\Omega)$ and in the weighted H\"older space
$C^0_s(\overline\Omega)$, respectively, do coincide.
