Regularity results for Choquard equations involving fractional $p$-Laplacian

Abstract

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.