We consider the fractional Laplacian with positive Dirichlet data          (−∆) α 2 u = λu p in Ω, u > 0 in Ω, u = ϕ in R n \Ω, where p > 1, 0 < α < min{2, n}, Ω ⊂ R n is a smooth bounded domain, ϕ is a nonnegative function, positive somewhere and satisfying some other conditions. We prove that there exists λ * > 0 such that for any 0 < λ < λ * , the problem admits at least one positive classical solution; for λ > λ * , the problem admits no classical solution. Moreover, for 1 < p ≤ n+α n−α , there exists 0 < λ ≤ λ * such that for any 0 < λ < λ, the problem admits a second positive classical solution. From the results obtained, we can see that the existence results of the fractional Laplacian with positive Dirichlet data are quite different from the fractional Laplacian with zero Dirichlet data.