On a super polyharmonic property of a higher-order fractional Laplacian

Abstract

Let 0 < α < 2, p ≥ 1, m ∞ ℕ₊. Consider the positive solution u of the PDE

$${(- \Delta)^{{\alpha \over 2} + m}}u(x) = {u^p}(x)\,\,\,{\rm{in}}\,\,{\mathbb{R}^n}.$$

((0.1))

In [1] (Transactions of the American Mathematical Society, 2021), Cao, Dai and Qin showed that, under the condition \(u \in {{\cal L}_\alpha}\), (0.1) possesses a super polyharmonic property \({(- \Delta)^{k + {\alpha \over 2}}}u \ge 0\) for k = 0,1, ⋯, m − 1. In this paper, we show another kind of super polyharmonic property (−Δ)^ku > 0 for k = 1, ⋯, m − 1, under the conditions \({(- \Delta)^m}u \in {{\cal L}_\alpha}\) and (−Δ)^mu ≥ 0. Both kinds of super polyharmonic properties can lead to an equivalence between (0.1) and the integral equation \(u(x) = \int_{{\mathbb{R}^n}} {{{{u^p}(y)} \over {|x - y{|^{n - 2m - \alpha}}}}{\rm{d}}y} \). One can classify solutions to (0.1) following the work of [2] and [3] by Chen, Li, Ou.