Existence of three positive solutions for a nonlocal singular dirichlet boundary problem

In this article, we prove the existence of at least three positive solutions for the following nonlocal singular problem \begin{equation*} (P_\la)\left{ \begin{split} (-\De)^su &= \la\frac{f(u)}{u^q}, ; ; u>0 ;; $\text{in}$;; \Om,\ u &= 0;; $\text{in}$;; \mb R^n \setminus \Om \end{split} \right. \end{equation*} where $(-\text{De}{)}^s$ denotes the fractional Laplace operator for $s\in (0,1)$, $n>2s$, $q\in (0,1)$, $\text{la}>0$ and $\text{Om}$ is smooth bounded domain in $\text{mb}{R}^n$. Here $f:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is a continuous nondecreasing map satisfying $\underset{u\to \mathrm{\infty }}{\lim }\frac{f(u)}{u^{q+1}}=0$. We show that under certain additional assumptions on $f$, $(P_{\text{la}})$ possesses at least three distinct solutions for a certain range of $\text{la}$. We use the method of sub-supersolutions and a critical point theorem by Amann \cite{amann} to prove our results. Moreover, we prove a new existence result for a suitable infinite semipositone nonlocal problem which played a crucial role to obtain our main result and is of independent interest. \medskip