Boundary value problems for Choquard equations

Abstract

We prove existence of a positive radial solution to the Choquard equation $$-\Delta u+Vu=(I_{\alpha }\ast |u{|}^p)|u{|}^{p-2}u\text{in}\Omega$$ 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.