Existence of groundstates for a class of nonlinear Choquard equations

We prove the existence of a nontrivial solution (u \in H^1 (\R^N)) to the nonlinear Choquard equation [- \Delta u + u = \bigl(I_\alpha \ast F (u)\bigr) F' (u) \quad \text{in (\R^N),}] where (I_\alpha) is a Riesz potential, under almost necessary conditions on the nonlinearity (F) in the spirit of Berestycki and Lions. This solution is a groundstate; if moreover (F) is even and monotone on ((0,\infty)), then (u) is of constant sign and radially symmetric.