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−Δpu−μ|x|pup−1=C|x|σuq in exterior domains of RNRN (N⩾2N⩾2). Here p∈(1,+∞)p∈(1,+∞) and μ⩽CHμ⩽CH, where CHCH is the critical Hardy constant. We provide a sharp characterization of the set of (q,σ)∈R2(q,σ)∈R2 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.