On a periodic Schrodinger equation with nonlocal superlinear part

We consider the Choquard-Pekar equation -Deltau+Vu=(W*u(2))u uis an element ofH(1) (R-3) and focus on the case of periodic potential V. For a large class of even functions W we show existence and multiplicity of solutions. Essentially the conditions are that 0 is not in the spectrum of the linear part -Delta+V and that W does not change sign. Our results carry over to more general nonlinear terms in arbitrary space dimension Ngreater than or equal to2.