Nonlinear Inequalities with Double Riesz Potentials

Abstract

We investigate the nonnegative solutions to the nonlinear integral inequality u ≥ Iα ∗((Iβ ∗ up)uq) a.e. in ${\mathbb R}^{N}$ ℝ N , where α, β ∈ (0, N), p, q > 0 and Iα, Iβ 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.

On Liouville theorems of a Hartree–Poisson system

“…Other results can refer to [20,22,43] and the references therein. Recently, Ghergu et al [12] shows a necessary and sufficient condition of existence of super-solutions of u(x) = |x| α−n * [u p−1 (|x| β−n * u p )], u > 0 on R n .…”

“…Recently, Ghergu et al. [12] shows a necessary and sufficient condition of existence of super-solutions of …”