preprint

Inserted: 17 may 2023
Last Updated: 17 may 2023

Year: 2023

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.