Positive solutions of the fractional Laplace equation with singular nonlinear terms

We study the existence of positive solutions for the fractional Laplace equation involving both singular and power-type nonlinearities: (−Δ)su=λf(x)uγ+M(x)upinΩ,u=0inRn∖Ω, where Ω⊂Rn is a bounded smooth domain, 0<s<1, γ>0, 1<p<2s∗−1, and f, M satisfy suitable assumptions. For 0<γ<1, we employ variational methods on the Nehari manifold to establish the existence of at least two distinct energy weak solutions for sufficiently small λ>0. For general γ>0, including the strong singular case γ≥1, we construct a weak solution via the method of sub- and supersolutions. Moreover, we discuss the regularity of the weak solution.