Our propose here is to provide a Hopf lemma and a strong minimum principle for weak supersolutions ofwhere Ω is an open set of R N , s ∈ (0, 1), p ∈ (1, +∞), c ∈ C(Ω) and (−∆ p ) s is the fractional p-Laplacian.

The purpose of this work is to study a finite element method for finding solutions to the eigenvalue problem for the fractional Laplacian. We prove that the discrete eigenvalue problem converges to the continuous one and we show the order of such convergence. Finally, we perform some numerical experiments and compare our results with previous work by other authors.2010 Mathematics Subject Classification. 46E35, 35P15, 49R05 65N25, 65N30.

Abstract. We prove the existence of an unbounded branch of solutions to the non-linear non-local equationbifurcating from the first eigenvalue. Here (−∆) s p denotes the fractional p-Laplacian and Ω ⊂ R n is a bounded regular domain. The proof of the bifurcation results relies in computing the Leray-Schauder degree by making an homotopy respect to s (the order of the fractional p-Laplacian) and then to use results of local case (that is s = 1) found in [17]. Finally, we give some application to an existence result.