An asymptotically periodic and asymptotically linear Schrödinger equation with indefinite linear part

We consider the semilinear Schrödinger equation {−△u+V(x)u=f(x,u),x∈RN,u∈H1(RN), where V(x) is asymptotically periodic and sign-changing, f(x,u) is asymptotically periodic in x, and asymptotically linear as |u|→∞. Under some weak assumptions on V and f, we prove that the above problem has a nontrivial solution.