Groundstates of the Choquard equations with a sign-changing self-interaction potential

We consider a nonlinear Choquard equation -Δu+u=(V∗|u|p)|u|p-2uinRN,when the self-interaction potential V is unbounded from below. Under some assumptions on V and on p, covering p= 2 and V being the one- or two-dimensional Newton kernel, we prove the existence of a nontrivial groundstate solution u∈ H1(RN) \ 0 by solving a relaxed problem by a constrained minimization and then proving the convergence of the relaxed solutions to a groundstate of the original equation.