Liouville Type Results For Polyharmonic Inequalities with Nonlocal Terms

Abstract

In this note, we study the polyharmonic inequalities system\[(-\Delta)^m u_i \geq \sum_{j=1}^n e_{i j}\left(\Psi_{i j}(|x|) * u_j^{p_{i j}}\right) u_i^{q_{i j}} \quad \text { in }{ }^N, \quad i=1,2, \cdots, n, \]where \(N \geq 1\) and \(m \geq 1\) are integers, \(p_{i j} \geq 1, q_{i j}>0\). \(\Delta^m\) denotes the m-polyharmonic operator. The operator \(*\) denotes the convolution and \(\Psi_{i j}\) is a function that has certain properties. \(\left(e_{i j}\right)\) is the adjacency matrix. By poly-superharmonic propery of u and some estimates, we get a Liouville type result of (0.1), which generalize the recent results on these inequalities.