On multiple solutions of a semilinear Schrödinger equation with periodic potential

This paper is concerned with the semilinear Schrödinger equation (S)−Δu+V(x)u=f(x,u),u∈H1(RN), where VV and ff are periodic in the xx-variables, ff is a superlinear and subcritical nonlinearity, and 0 lies in a spectral gap of −Δu+V−Δu+V. It is shown that, if ff is odd in uu then (S) has infinitely many large energy solutions. The proof relies on a generalized variant fountain theorem for strongly indefinite functionals, established in this paper.