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

We consider a nonlinear Choquard equation $$-\mathrm{\Delta }u+u=(V\ast \mathrm{\mid }u{\mathrm{\mid }}^p)\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u\text{in }{\mathbb{R}}^N,$$ 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 $u\in H^1({\mathbb{R}}^N)\setminus \{0\}$ 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.