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

Abstract

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,where0<α<2, σ1, σ2∈(-1,1), Dis a boundedC1,1-domain inℝn,n≥2,andδ(x)denotes the Euclidian distance fromxto the boundary ofD.The nonnegative weight functionsa1, a2are required to satisfy certain hypotheses related to the Karamata class. We also investigate the global behavior of such solution.