A pr 2 00 6 Positive solutions to nonlinear p - Laplace equations with Hardy potential in exterior domains

We study the existence and nonexistence of positive (super) solutions to the nonlinear p-Laplace equation −∆pu− μ |x|p u = C |x|σ u in exterior domains of R (N ≥ 2). Here p ∈ (1,+∞) and μ ≤ CH , where CH is the critical Hardy constant. We provide a sharp characterization of the set of (q, σ) ∈ R such that the equation has no positive (super) solutions. 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 p-Laplace operator with Hardy-type potentials, comparison principles and an improved version of Hardy’s inequality in exterior domains. In the context of the p-Laplacian we establish the existence and asymptotic behavior of the harmonic functions by means of the generalized Prüfer-Transformation.