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

Abstract: We consider a nonlinear Choquard equation $$ -\Delta u+u= (V * |u|^p )|u|^{p-2}u \qquad \text{in }\mathbb{R}^N, $$ 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.