Nonlinear Inequalities with Double Riesz Potentials

Abstract

We investigate the nonnegative solutions to the nonlinear integral inequality $u\ge I_{\alpha }\ast ((I_{\beta }\ast u^p)u^q)$ a.e. in ${\mathbb{R}}^N$, where $\alpha ,\beta \in (0,N)$, $p,q>0$ and $I_{\alpha }$, $I_{\beta }$ denote the Riesz potentials of order $\alpha$ and $\beta$ respectively. Our approach relies on a nonlocal positivity principle which allows us to derive optimal ranges for the parameters $\alpha$, $\beta$, $p$ and $q$ to describe the existence and the nonexistence of a solution. The optimal decay at infinity for such solutions is also discussed.

15 pages

References in corpus (3)

• Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains
• Qualitative properties of solutions for an integral system related to the Hardy-Sobolev inequality
• Nonlocal Hardy type inequalities with optimal constants and remainder terms