  We study the existence and non-existence of classical solutions for
inequalities of type $$ \pm \Delta^m u \geq \big(\Psi(|x|)*u^p\big)u^q
\quad\mbox{ in } {\mathbb R}^N (N\geq 1). $$ Here, $\Delta^m$ $(m\geq 1)$ is
the polyharmonic operator, $p, q>0$ and $*$ denotes the convolution operator,
where $\Psi>0$ 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.
