Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms

Based on new information concerning strongly indefinite functionals without Palais–Smale conditions, we study existence and multiplicity of solutions of the Schrödinger equation where V and g are periodic with respect to x and 0 lies in a gap of σ(-Δ+V). Supposing g is asymptotically linear as u→∞ and symmetric in u, we obtain infinitely many geometrically distinct solutions. We also consider the situation where g is super linear with mild assumptions different from those studied previously, and establish the existence and multiplicity.