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 equationin 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.

Bessel pairs and optimal Hardy and Hardy–Rellich inequalities

“…It can also be used to improve the recent results of Liskevich et al [23] on exterior domains and will be developed in a forthcoming paper.…”

Existence and Liouville-type theorems for some indefinite quasilinear elliptic problems with potentials vanishing at infinity

“…Thus, by combining (44) with (46), we find out that for some constant ξ > 0, depending only on p, q, s, Λ 1 and Γ ,…”

“…where L stands here for some second-order elliptic operator, specified in the study, while Ω is either the entire R N , or a cone, or an exterior domain; we refer to the works [7,8,10,16,17,20,24,25,27,29,31,35,39,41,42,58,60,61,64,68] in which L coincides with the standard Laplacian, to [11,12,14,19,21,44,47,63] where L is its nonlinear counterpart, the p-Laplacian, and to [13,[34][35][36]43,45,48,50,54,55] where more general linear or nonlinear elliptic operators are considered. Nevertheless, the above list is by no means exhaustive and the reader who wishes to get a panoramic view of this fascinating field should also consult the extensive treatise [49], as well as the very recent surveys [28] and [37].…”

“…[10,16,17,20,23,24,26,31,39,58]), as well as on the derivation of pointwise estimates like Phragmén-Lindelöf-type bounds or Harnacktype estimates (cf. [34][35][36][43][44][45]), or (iii) pointwise estimates via nonlinear potential theory (cf. [54,55]).…”

Hardy’s Inequality for Functions Vanishing on a Part of the Boundary

“…Moreover, over the last years it became manifest that Hardy's inequality plays an eminent role in modern PDE theory, see e.g. [7,46,43,2,13,9,16,23,32,34].…”