Multiplicity and Hölder regularity of solutions for a nonlocal elliptic PDE involving singularity

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} (-Δ_p)^s u&=& \fracλ{u^γ}+u^q~\text{in}~Ω,\nonumber u&=&0~\text{in}~\mathbb{R}^N\setminusΩ,\nonumber u&>& 0~\text{in}~Ω\nonumber, \end{eqnarray} where $Ω$ is an open bounded domain in $\mathbb{R}^N$ with smooth boundary, $N>ps$, $s\in (0,1)$, $λ>0$, $0<γ<1$, $1<p<\infty$, $p-1<q\leq p_s^{*}=\frac{Np}{N-ps}$. We employ variational techniques to show the existence of multiple positive weak solutions of the above problem. We also prove that for some $β\in (0,1)$, the weak solution to the problem is in $C^{1,β}(\overlineΩ)$.