Elliptic inequalities with nonlinear convolution and Hardy terms in cone-like domains

Abstract

We study the inequality −Δu−μ|x|2u≥(|x|−α⁎up)uq in an unbounded cone CΩρ⊂RN (N≥2) generated by a subdomain Ω of the unit sphere SN−1⊂RN, p,q,ρ>0, μ∈R and 0≤α<N. In the above, |x|−α⁎up denotes the standard convolution operator in the cone CΩρ. We discuss the existence and nonexistence of positive solutions in terms of N,p,q,α,μ and Ω. Extensions to systems of inequalities are also investigated.