Boundary value problems for Choquard equations

We consider the following nonlinear Choquard equation −Δu+Vu=(Iα∗|u|p)|u|p−2uinΩ⊂RN, where N≥2, p∈(1,+∞), V(x) is a continuous radial function such that infx∈ΩV>0 and Iα(x) is the Riesz potential of order α∈(0,N). Assuming Neumann or Dirichlet boundary conditions, we prove existence of a positive radial solution to the corresponding boundary value problem when Ω is an annulus, or an exterior domain of the form RN∖Br(0)¯. We also provide a nonexistence result: if p≥[Formula presented] the corresponding Dirichlet problem has no nontrivial regular solution in strictly star-shaped domains. Finally, when considering annular domains, letting α→0+ we recover existence results for the corresponding local problem with power-type nonlinearity.