Abstract The aim of this paper is to investigate the existence of solutions of the non-local elliptic problem { ( - Δ ) s u = | u | p - 2 u + h ( x ) in Ω , u = 0 on ℝ n ∖ Ω , \left\{\begin{aligned} &\displaystyle(-\Delta)^{s}u\ =\lvert u\rvert^{p-2}u+h(% x)&&\displaystyle\text{in }\Omega,\\ &\displaystyle{u=0}&&\displaystyle\text{on }\mathbb{R}^{n}\setminus\Omega,\end% {aligned}\right. where s ∈ ( 0 , 1 ) {s\in(0,1)} , n > 2 s {n>2s} , Ω is an open bounded domain of ℝ n {\mathbb{R}^{n}} with Lipschitz boundary ∂ Ω {\partial\Omega} , ( - Δ ) s {(-\Delta)^{s}} is the non-local Laplacian operator, 2 < p < 2 s ∗ {2<p<2_{s}^{\ast}} and h ∈ L 2 ( Ω ) {h\in L^{2}(\Omega)} . This problem requires the study of the eigenvalue problem related to the fractional Laplace operator, with or without potential.