EXISTENCE OF GROUNDSTATES FOR A CLASS OF NONLINEAR CHOQUARD EQUATIONS

We prove the existence of a nontrivial solution u is an element of H-1(R-N) to the nonlinear Choquard equation Delta u + u = (I-alpha * F(u)) F'(u) 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 and has additional local regularity properties; if moreover F is even and monotone on (0, infinity), then u is of constant sign and radially symmetric.