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:\left\{\begin{aligned} \displaystyle(-\Delta)^{s}u&\displaystyle=\lambda\frac{% f(u)}{u^{q}},&&\displaystyle u>0\text{ in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{in }\mathbb{R}^{% n}\setminus\Omega,\end{aligned}\right.where {(-\Delta)^{s}} denotes the fractional Laplace operator for {s\in(0,1)}, {n>2s}, {q\in(0,1)}, {\lambda>0} and Ω is a smooth bounded domain in {\mathbb{R}^{n}}. Here {f:[0,\infty)\to[0,\infty)} is a continuous nondecreasing map satisfying\lim_{u\to\infty}\frac{f(u)}{u^{q+1}}=0.We show that under certain additional assumptions on f, the above problem possesses at least three distinct solutions for a certain range of λ. We use the method of sub-supersolutions and a critical point theorem by Amann [H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 1976, 4, 620–709] 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.