POSITIVE SOLUTIONS TO SINGULAR SEMILINEAR ELLIPTIC EQUATIONS WITH CRITICAL POTENTIAL ON CONE-LIKE DOMAINS

We study the existence and nonexistence of positive (super-) solutions to a singular semilinear elliptic equation -del.(vertical bar x vertical bar(A)del u) - B vertical bar x vertical bar(A-2)u = C vertical bar x vertical bar(A-sigma)u(p) in cone-like domains of R-N (N >= 2), for the full range of parameters A, B, sigma, p is an element of R and C > 0. We provide a characterization of the set of (p, sigma) is an element of R-2 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, Phragmen-Lindelof type comparison arguments and an improved version of Hardy's inequality in cone-like domains.