Positive solutions to singular semilinear elliptic equations with critical potential on cone-like domains

Abstract

We study the existence and nonexistence of positive (super-) solutions to a singular semilinear elliptic equation -∇.(|x|A ∇u) -B|x|A-2u = C|x|A-σ up in cone-like domains of RN (N ≥ 2), for the full range of parameters A, B,σ, p ∈ R and C > 0. We provide a characterization of the set of (p, σ) ∈ R2 such that the equation has no positive (super-) solutions, depending on the values of A, B and the principal Dirichlet eigenvalue of the cross-section of the cone. The proofs are based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the Laplace operator with critical potentials, Phragmén-Lindelöf type comparison arguments and an improved version of Hardy's inequality in cone-like domains.