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The goal of this paper is to study the following non-local superlinear elliptic problem (-Δ)su=|u|p-σϕ1inΩ,u=0inRN\Ω,u>0inΩ,where (-Δ)s is the fractional Laplace operator, Ω⊂RN is an open domain with Lipschitz boundary, σ>0, p∈(1,2s∗-1) with 2s∗=2NN-2s and ϕ1 is the first positive eigenfunction of the fractional Laplacian with Dirichlet boundary condition. We prove the non-local version of a conjecture due to Lazer and McKenna (Proc R Soc Edinb Sect A 95:275-283, 1983) by constructing solutions with sharp peaks near local maximum points of ϕ1. [ABSTRACT FROM AUTHOR]