Boundary value problems for Choquard equations

We consider the following nonlinear Choquard equation -+=(*||)||-2 in subset of R, where N >= 2, pe (1,400), V(x) is a continuous radial function such that inf0 and (4) is the Riesz potential of order a (0, N). Assuming Neumann Dirichlet boundary conditions, we prove existence of a positive radial solution to the corresponding boundary value problem when 2 is an annulus, or an exterior domain of the form R8,(0). We also provide a nonexistence result: if the corresponding Dirichlet problem has no nontrivial regular solution in strictly star-shaped domains. Finally, when considering annular domains, letting 0 we recover existence results for the corresponding local problem with power-type nonlinearity.