Representation formulae for solutions to some classes of higher order systems and related Liouville theorems
type: Item
instance of: scholarly article
title: Representation formulae for solutions to some classes of higher order systems and related Liouville theorems
zbMATH Open document ID: 1186.35026
author: Gabriella Caristi
author: Lorenzo D'Ambrosio
author: Enzo Mitidieri
published in: Milan Journal of Mathematics
publication date: 2009-11-02T00:00:00Z
review text: The authors studied some representation formulae and related Liouville theorems for solutions to some classes of higher order differential equations and systems on Euclidean space. Let \(m \geq 1\) be an integer and \(N > 2m\). Let \(\mu\) be a positive Radon measure on \({\mathbb R}^N\) and \(l \in {\mathbb R}\). Then the authors proved that if \(u\) is a solution of \((-\Delta)^m u = \mu\) on \({\mathbb R}^N\) in the distribution sense and for a.e. \(x \in {\mathbb R}^N\), \[ \liminf_{R\to \infty}\frac{1}{R^N}\int_{R\leq |x-y|\leq 2R} |u(y) - l| \,dy = 0,\tag{1} \] then \(u \in L^l_{\text{loc}}({\mathbb R}^N)\) and \[ u(x) = l + C \int_{{\mathbb R}^N} \frac{d\mu(y)}{|x-y|^{N-2m}}\quad \text{a.e. } x\in {\mathbb R}^N,\tag{2} \] where \(C\) is a positive constant depending only on \(m\) and \(N\). The converse is also true. As a corollary, one can replace the positive Radon measure \(\mu\) by a function \(h \in L^l_{\text{loc}}({\mathbb R}^N)\). Namely if \(u\) is a distributional solution of \((-\Delta)^m u \geq h\) and (1) holds, then \(u\) has of the form (2) when \(d\mu\) is replaced by \(h\). As a consequence of the representation formula (2), the authors obtained some Liouville theorems. For instance, let \(p >1\) and let \(u \in L^1_{\text{loc}}({\mathbb R}^N)\) be a distributional solution of \((-\Delta)^m u \geq 0\) with \(N > 2m\). If \(u \in L^p({\mathbb R}^N)\) with \((N-2m)p \leq N\) or \(u \in L^p_\omega({\mathbb R}^N)\), the weak-\(L^p\) space, with \((N-2m)p < N\), then \(u \equiv 0\) a.e. on \({\mathbb R}^N\). Next, the authors obtained some representation formulae and related Liouville theorems for polyharmonic systems related with the Hardy-Littlewood-Sobolev system (HLS for brevity). Using their representation formulae for polyharmonic systems, the authors proved a non-existence result for non-negative radial solutions of the HLS systems of integral equations.
reviewed by: Gabjin Yun
Mathematics Subject Classification ID: 35C15
Mathematics Subject Classification ID: 31C10
Mathematics Subject Classification ID: 45G15
Mathematics Subject Classification ID: 35B53
Mathematics Subject Classification ID: 35J30
zbMATH DE Number: 5624119
zbMATH Keywords: Hardy-Littlewood-Sobolev inequality
zbMATH Keywords: polyharmonic systems
zbMATH Keywords: Hardy-Littlewood-Sobolev system
MaRDI profile type: Publication
full work available at URL: s00032-008-0090-3
OpenAlex ID: W2145543724
DOI: 10.1007/S00032-008-0090-3
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description: scientific article; zbMATH DE number 5624119