On some critical problems for the fractional Laplacian operator

We study the effect of lower order perturbations in the existence of positive solutions to the following critical elliptic problem involving the fractional Laplacian:{(−Δ)α/2u=λuq+uN+αN−α,u>0in Ω,u=0on ∂Ω, where Ω⊂RNΩ⊂RN is a smooth bounded domain, N⩾1N⩾1, λ>0λ>0, 00Λ>0, at least one if λ=Λλ=Λ, no solution if λ>Λλ>Λ. For q=1q=1 we show existence of at least one solution for 0<λ<λ10<λ<λ1 and nonexistence for λ⩾λ1λ⩾λ1. When q>1q>1 the existence is shown for every λ>0λ>0. Also we prove that the solutions are bounded and regular.