  In this article, first we address the regularity of weak solution for a class
of $p$-fractional Choquard equations:
  \begin{equation*}
  \;\;\; \left.\begin{array}{rl}
  (-\Delta)_p^su&=\left(\displaystyle\int_\Omega\frac{F(y,u)}{|x-y|^{\mu}}dy\right)f(x,u),\hspace{5mm}x\in
\Omega,
  u&=0,\hspace{35mm}x\in \mathbb R^N\setminus \Omega,
  \end{array}
  \right\}
  \end{equation*}
  where $\Omega\subset\mathbb R^N$ is a smooth bounded domain, $1<p<\infty$ and
$0<s<1$ such that $sp<N,$ $0<\mu<\min\{N,2sp\}$ and $f:\Omega\times\mathbb
R\to\mathbb R$ is a continuous function with at most critical growth condition
(in the sense of Hardy-Littlewood-Sobolev inequality) and $F$ is its primitive.
Next, for $p\geq2,$ we discuss the Sobolev versus H\"{o}lder minimizers of the
energy functional $J$ associated to the above problem, and using that we
establish the existence of the local minimizer of $J$ in the fractional Sobolev
space $W_0^{s,p}(\Omega).$ Moreover, we discuss the aforementioned results by
adding a local perturbation term (at most critical in the sense of Sobolev
inequality) in the right-hand side in the above equation.
