Polyharmonic inequalities with nonlocal terms

Abstract

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

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