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

We consider a nonlinear Choquard equation when the self-interaction potential V is unbounded from below. Under some assumptions on and on , covering and being the one- or two-dimensional Newton kernel, we prove the existence of a nontrivial groundstate solution 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.