Nonlinear Inequalities with Double Riesz Potentials

Abstract

We investigate the nonnegative solutions to the nonlinear integral inequality u ≥ Iα * (I β * u p )u q a.e. in R N , where α, β ∈ (0, N ), p, q > 0 and Iα, 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.