Boundary value problems for Choquard equations

TL;DR

This paper establishes the existence of positive radial solutions for boundary value problems involving the Choquard equation in certain domains, and also identifies conditions under which solutions do not exist.

Contribution

It proves existence of solutions in annular and exterior domains, and provides nonexistence results for certain parameter ranges and domain shapes.

Findings

Existence of positive radial solutions in annular and exterior domains.

Nonexistence of solutions for p ≥ (N+α)/(N-2) in star-shaped domains.

Conditions under which boundary value problems for Choquard equations have solutions.

Abstract

We prove existence of a positive radial solution to the Choquard equation $$-\Delta u +V u=(I_\alpha\ast |u|^p)|u|^{p-2}u\qquad\text{in}\,\,\,\Omega$$ with Neumann or Dirichlet boundary conditions, when $\Omega$ is an annulus, or an exterior domain of the form $\mathbb{R}^N\setminus \bar{B}_a(0)$. We provide also a nonexistence result, that is if $p\ge\frac{N+\alpha}{N-2}$ the corresponding Dirichlet problem does not have any nontrivial regular solution in strictly strictly star-shaped domains.