Positive solutions to singular semilinear elliptic equations with critical potential on cone–like domains
نویسندگان
چکیده
We study the existence and nonexistence of positive (super-) solutions to a singular semilinear elliptic equation −∇ · (|x|∇u)−B|x|u = C|x|u in cone–like domains of R (N ≥ 2), for the full range of parameters A,B, σ, p ∈ R and C > 0. We provide a complete characterization of the set of (p, σ) ∈ R such that the equation has no positive (super-) solutions, depending on the values of A,B and the principle 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–Lindelöf type comparison arguments and an improved version of Hardy’s inequality in cone–like domains.
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