Existence of Three Positive Solutions for a Nonlocal Singular Dirichlet Boundary Problem

Abstract

Abstract In this article, we prove the existence of at least three positive solutions for the following nonlocal singular problem: { ( - Δ ) s ⁢ u = λ ⁢ f ⁢ ( u ) u q , u > 0 ⁢  in  ⁢ Ω , u = 0 in  ⁢ ℝ n ∖ Ω , \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 ( - Δ ) s {(-\Delta)^{s}} denotes the fractional Laplace operator for s ∈ ( 0 , 1 ) {s\in(0,1)} , n > 2 ⁢ s {n>2s} , q ∈ ( 0 , 1 ) {q\in(0,1)} , λ > 0 {\lambda>0} and Ω is a smooth bounded domain in ℝ n {\mathbb{R}^{n}} . Here f : [ 0 , ∞ ) → [ 0 , ∞ ) {f:[0,\infty)\to[0,\infty)} is a continuous nondecreasing map satisfying lim u → ∞ ⁡ f ⁢ ( u ) u q + 1 = 0 . \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.