Non-homogeneous fourth order elliptic inequalities with the convolution term

Abstract

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