Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains

TL;DR

This paper investigates the existence and decay properties of supersolutions to a class of nonlinear elliptic equations, including the Choquard equation, in exterior domains, establishing nonexistence results and optimal decay rates.

Contribution

It provides new nonexistence results and decay rate characterizations for supersolutions of Choquard-type equations in exterior domains.

Findings

Certain parameter ranges admit no nontrivial supersolutions.

When supersolutions exist, their decay rates are optimal.

The techniques apply to a broad class of nonlinear elliptic problems.

Abstract

We consider a semilinear elliptic problem with a nonlinear term which is the product of a power and the Riesz potential of a power. This family of equations includes the Choquard or nonlinear Schroedinger--Newton equation. We show that for some values of the parameters the equation does not have nontrivial nonnegative supersolutions in exterior domains. The same techniques yield optimal decay rates when supersolutions exists.