Non-homogeneous fourth order elliptic inequalities with the convolution term

We are concerned with the study of the twin non-local inequalities featuring non-homogeneous differential operators $${\displaystyle -{\mathrm{\Delta }}^{2}u+\lambda \mathrm{\Delta }u\ge (K_{\alpha ,\beta }\ast u^p)u^q\text{ in }{\mathbb{R}}^N(N\ge 1),}$$ and $${\displaystyle {\mathrm{\Delta }}^{2}u-\lambda \mathrm{\Delta }u\ge (K_{\alpha ,\beta }\ast u^p)u^q\text{ in }{\mathbb{R}}^N(N\ge 1),}$$ with parameters $\lambda ,p,q>0$, $0\le \alpha \le N$ and $\beta >\alpha -N$. In the above inequalities the potential $K_{\alpha ,\beta }$ is given by $K_{\alpha ,\beta }(x)=\mathrm{\mid }x{\mathrm{\mid }}^{-\alpha }{\log }^{\beta }(1+\mathrm{\mid }x\mathrm{\mid })$ while $K_{\alpha ,\beta }\ast u^p$ denotes the standard convolution operator in ${\mathbb{R}}^N$. We discuss the existence and non-existence of non-negative solutions in terms of $N,p,q,\lambda ,\alpha$ and $\beta$.