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We consider a class of generalized nonlocal $p$
-Laplacian equations. We find some proper structural conditions to establish a version of nonlocal Harnack inequalities of weak solutions to such nonlocal problems by using the expansion of positivity and energy estimates.
We prove the partial Hölder continuity for minimizers of quasiconvex functionals
where 
\begin{equation*} \mathcal{F}({\bf u}) \colon =\int_{\Omega} f(x,{\bf u},D{\bf u})\,\textrm{d} x, \end{equation*}
$f$ satisfies a uniform VMO condition with respect to the 
$x$-variable and is continuous with respect to 
${\bf u}$. The growth condition with respect to the gradient variable is assumed a general one.
