Existence and multiplicity of positive solutions for the fractional Laplacian under subcritical or critical growth

We study a Dirichlet type problem for an equation involving the fractional Laplacian and a reaction term subject to either subcritical or critical growth conditions, depending on a positive parameter. Applying a critical point result of Bonanno, we prove existence of one or two positive solutions as soon as the parameter lies under an (explicitly determined) value. As an application, we find two positive solutions for a fractional Ambrosetti–Brezis–Cerami problem.