Existence and Uniqueness of Positive Solution for a Fractional Dirichlet Problem with Combined Nonlinear Effects in Bounded Domains

We prove the existence and uniqueness of a positive continuous solution to the following singular semilinear fractional Dirichlet problem (-Δ)α/2u=a1(x)uσ1+a2(x)uσ2, in  D  limx→z∈∂D(δ(x))1-(α/2)u(x)=0, where 0<α<2, σ1,  σ2∈(-1,1), D is a bounded C1,1-domain in ℝn,n≥2, and δ(x) denotes the Euclidian distance from x to the boundary of D. The nonnegative weight functions a1,  a2 are required to satisfy certain hypotheses related to the Karamata class. We also investigate the global behavior of such solution