Choquard equations under confining external potentials

We consider the nonlinear Choquard equation $$-\Delta u+V u=(I_\alpha \ast \vert u\vert ^p)\vert u\vert ^{p-2}u \qquad \text{ in } \mathbb{R}^N$$ where $N\geq 1$, $I_\alpha$ is the Riesz potential integral operator of order $\alpha \in (0, N)$ and $p > 1$. If the potential $V \in C (\mathbb{R}^N; [0,+\infty))$ satisfies the confining condition $$\liminf\limits_{\vert x\vert \to +\infty}\frac{V(x)}{1+\vert x\vert ^{\frac{N+\alpha}{p}-N}}=+\infty,$$ and $\frac{1}{p} > \frac{N - 2}{N + \alpha}$, we show the existence of a groundstate, of an infinite sequence of solutions of unbounded energy and, when $p \ge 2$ the existence of least energy nodal solution. The constructions are based on suitable weighted compact embedding theorems. The growth assumption is sharp in view of a Poho\v{z}aev identity that we establish.