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

We consider the semilinear Schrödinger equation { − △ u + V ( x ) u = f ( x , u ) , for x ∈ R N , 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 . 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.