Title: On some critical problems for the fractional Laplacian operator
Abstract: 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⊂RN is a smooth bounded domain, N⩾1, λ>0, 0<q<N+αN−α, 0<α<min{N,2}. For suitable conditions on α depending on q, we prove: In the case q<1, there exist at least two solutions for every 0<λ<Λ and some Λ>0, at least one if λ=Λ, no solution if λ>Λ. For q=1 we show existence of at least one solution for 0<λ<λ1 and nonexistence for λ⩾λ1. When q>1 the existence is shown for every λ>0. Also we prove that the solutions are bounded and regular.