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

TL;DR

This paper investigates the regularity of solutions for fractional p-Laplacian Choquard equations, compares Sobolev and H"older minimizers, and explores the effects of perturbations on these solutions.

Contribution

It provides new regularity results, analyzes minimizers in Sobolev and H"older spaces, and studies the impact of perturbations on the solutions of fractional Choquard equations.

Findings

Established regularity of weak solutions.

Compared Sobolev and H"older minimizers.

Analyzed effects of perturbations on solutions.

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.