Positive solutions to nonlinear p-Laplace equations with Hardy potential in exterior domains

Abstract

We study the existence and nonexistence of positive (super)solutions to the nonlinear p-Laplace equation − Δ p u − μ | x | p u p − 1 = C | x | σ u q in exterior domains of R N ( N ⩾ 2 ). Here p ∈ ( 1 , + ∞ ) and μ ⩽ C H , where C H is the critical Hardy constant. We provide a sharp characterization of the set of ( q , σ ) ∈ R 2 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.