A critical fractional equation with concave–convex power nonlinearities

Abstract

In this work we study the following fractional critical problem (Pλ)={(−Δ)su=λuq+u2s⁎−1,u>0inΩ,u=0inRn∖Ω, where Ω⊂Rn is a regular bounded domain, λ>0, 0<s<1 and n>2s. Here (−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) for different values of λ. The dependency on this parameter changes according to whether we consider the concave power case ( 0<q<1) or the convex power case ( 1<q<2s⁎−1). These two cases will be treated separately.