  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.

Analysis of PDEs