New conditions on nonlinearity for a periodic Schrödinger equation having zero as spectrum

We consider the semilinear Schrödinger equation{−△u+V(x)u=f(x,u),for x∈RN,u(x)→0,as |x|→∞, where f is a superlinear and subcritical nonlinearity. We mainly study the case when both V and f are periodic in x   and 0 is a boundary point of a spectral gap of −△+V−△+V. We extend a linking theorem of Kryszewski and Szulkin [15] and establish a new variational setting which is more suitable to the above case. We obtain two theorems on the existence of ground state solutions with mild assumptions on f.