Representation formulae for solutions to some classes of higher order systems and related Liouville theorems. (English) Zbl 1186.35026

Caristi, Gabriella; D’Ambrosio, Lorenzo; Mitidieri, Enzo

Milan J. Math. 76, 27-67 (2008).

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.

Reviewer: Gabjin Yun (Yongin)

MSC:

35C15	Integral representations of solutions to PDEs
31C10	Pluriharmonic and plurisubharmonic functions
45G15	Systems of nonlinear integral equations
35B53	Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs
35J30	Higher-order elliptic equations

Keywords:

Hardy-Littlewood-Sobolev inequality; polyharmonic systems; Hardy-Littlewood-Sobolev system