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

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In this article, we prove the existence of at least three positive solutions for the following nonlocal singular problem equation* (P){ split (-)ˢu &= (u)/uq, u>0 in, u &= 0 in Rⁿ split. equation* where (-)ˢ denotes the fractional Laplace operator for s∈ (0,1), n>2s, q ∈ (0,1), >0 and is smooth bounded domain in Rⁿ. Here f:[0,∞) → [0,∞) is a continuous nondecreasing map satisfying ᵤ→ ∞f(u)uq⁺¹=0. We show that under certain additional assumptions on f, (P) possesses at least three distinct solutions for a certain range of. We use the method of sub-supersolutions and a critical point theorem by Amann 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.