Multiplicity and H\"{o}lder regularity of solutions for a nonlocal elliptic PDE involving singularity

TL;DR

This paper establishes the existence of multiple positive solutions for a nonlinear nonlocal elliptic PDE with singularity, and demonstrates their Hölder regularity, using variational methods in a bounded domain.

Contribution

It introduces new results on multiple solutions and Hölder regularity for a nonlocal PDE with singularity, expanding understanding of such equations.

Findings

Multiple positive weak solutions exist for the PDE.

Solutions possess Hölder continuous regularity in the domain.

Variational techniques effectively prove existence and regularity.

Abstract

In this paper, we prove the existence of multiple solutions for a nonlinear nonlocal elliptic PDE involving a singularity which is given as \begin{eqnarray} (-\Delta_p)^s u&=& \frac{\lambda}{u^\gamma}+u^q~\text{in}~\Omega,\nonumber u&=&0~\text{in}~\mathbb{R}^N\setminus\Omega,\nonumber u&>& 0~\text{in}~\Omega\nonumber, \end{eqnarray} where $\Omega$ is an open bounded domain in $\mathbb{R}^N$ with smooth boundary, $N>ps$, $s\in (0,1)$, $\lambda>0$, $0<\gamma<1$, $1<p<\infty$, $p-1<q\leq p_s^{*}=\frac{Np}{N-ps}$. We employ variational techniques to show the existence of multiple positive weak solutions of the above problem. We also prove that for some $\beta\in (0,1)$, the weak solution to the problem is in $C^{1,\beta}(\overline{\Omega})$.