We study the existence and non-existence of classical solutions for inequalities of type$\pm {\mathrm{\Delta }}^{m}u\ge (\mathrm{\Psi }(|x|)⁎u^p)u^q\text{ in }{\mathbb{R}}^N(N\ge 1).$ Here, ${\mathrm{\Delta }}^m$ $(m\ge 1)$ is the polyharmonic operator, $p,q>0$ and ⁎ denotes the convolution operator, where $\mathrm{\Psi }>0$ is a continuous non-increasing function. We devise new methods to deduce that solutions of the above inequalities satisfy the poly-superharmonic property. This further allows us to obtain various Liouville type results. Our study is also extended to the case of systems of simultaneous inequalities.

中文翻译：

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具有非局部项的多谐不等式

我们研究了类型不等式的经典解的存在和不存在$\pm {\mathrm{\Delta }}^{米}你\ge (\mathrm{\Psi }(|X|)⁎{你}^{磷}){你}^q\text{ 在 }{\mathbb{电阻}}^N(N\ge 1).$ 这里， ${\mathrm{\Delta }}^{米}$ $(米\ge 1)$ 是多谐算子， $磷,q>0$ 和 ⁎ 表示卷积算子，其中 $\mathrm{\Psi }>0$是一个连续的非递增函数。我们设计了新方法来推断上述不等式的解满足多超谐波性质。这进一步使我们能够获得各种 Liouville 类型的结果。我们的研究还扩展到同时不等式系统的情况。