Hardy inequalities resulted from nonlinear problems dealing with A-Laplacian

Abstract

Abstract. We derive Hardy inequalities in weighted Sobolev spaces via anticoercive partial differential inequalities of elliptic type involving ALaplacian −ΔAu = −divA(∇u) ≥ Φ, where Φ is a given locally integrable function and u is defined on an open subset Ω ⊆ R n . Knowing solutions we derive Caccioppoli inequalities for u. As a consequence we obtain Hardy inequalities for compactly supported Lipschitz functions involving certain measures, having the formwhereĀ(t) is a Young function related to A and satisfying Δ -condition, while FĀ(t) = 1/(Ā(1/t)). Examples involvingĀ(t) = t p log α (2+t), p ≥ 1, α ≥ 0 are given. The work extends our previous work (Skrzypczaki, in Nonlinear Anal TMA 93:30-50, 2013), where we dealt with inequality −Δpu ≥ Φ, leading to Hardy and Hardy-Poincaré inequalities with the best constants. Mathematics Subject Classification (2010). Primary 26D10; Secondary 31B05, 35D30, 35J60, 35R45.

Caccioppoli-type estimates and Hardy-type inequalities derived from weighted p-harmonic problems

“…These cases are recalled in Remark 4.1. Furthermore, we provide here also new inequalities with the optimal constants, see The approach presented here and in the papers [55,56] is a modification of techniques originating in [43]. In all of these papers, the investigations start with derivation of Caccioppoli-type estimates for the solutions to nonlinear problems.…”

“…Other interesting results linking the existence of solutions in elliptic and parabolic PDEs with Hardy type inequalities are presented in [2,4,36,61,62], see also references therein. We refer also to the contribution by the third author [56], where instead of the nonweighted p-Laplacian in (1.1) one deals with the A-Laplacian: A u = div A (|∇u|) |∇u| 2 ∇u , involving a function A from an Orlicz class. Similar estimates in the framework of nonlocal operators can be found e.g.…”

Caccioppoli--type estimates and Hardy--type inequalities derived from degenerated $p$--harmonic problems

“…The approach presented here and in the papers [40] and [42] is the modification of methods from [25]. In all of these papers, the investigations start with derivation of Caccioppoli-type estimates for the solutions to nonlinear problem.…”

“…Other interesting results linking the existence of solutions in elliptic and parabolic PDEs with Hardy type inequalities are presented in [2,4,22,45,46], see also references therein. We refer also to the recent contribution by the third author [42], where, instead of the nondegenerated p-Laplacian in (1.1), one deals with the A-Laplacian:…”

Variable exponent Hardy-type inequalities in $\mathbb{R}^n$

“…Let us mention such branches as functional analysis, harmonic analysis, probability theory, and PDEs. Weighted versions of Hardy-type inequalities are also investigated on their own in the classical way [17,22,27], as well as in the various generalised frameworks [4,5,14,29].…”