Nonlinear Inequalities with Double Riesz Potentials

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

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