A critical fractional equation with concave–convex power nonlinearities

In this work we study the following fractional critical problem(Pλ)={(−Δ)su=λuq+u2s⁎−1,u>0in Ω,u=0in Rn∖Ω, where Ω⊂RnΩ⊂Rn is a regular bounded domain, λ>0λ>0, 02sn>2s. Here (−Δ)s(−Δ)s denotes the fractional Laplace operator defined, up to a normalization factor, by−(−Δ)su(x)=∫Rnu(x+y)+u(x−y)−2u(x)|y|n+2sdy,x∈Rn. Our main results show the existence and multiplicity of solutions to problem (Pλ)(Pλ) for different values of λ  . The dependency on this parameter changes according to whether we consider the concave power case (0