Infinitely many solutions for a Schrödinger equation with sign-changing potential and nonlinear term

We propose a new variational approach to find multiple critical points for strongly indefinite variational problems without assuming the weak upper semicontinuity on the variational functionals. By this approach, we obtain the existence of infinitely many geometrically distinct solutions for a stationary periodic Schrödinger equation, in which the linear part is strongly indefinite and the nonlinear term is allowed to change sign in general ways.