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-Δu+V(x)u=g(x,u)forx∈RN,u(x)→0as|x|→∞,where VV and gg are periodic with respect to xx and 00 lies in a gap of σ(-Δ+V)σ(-Δ+V). Supposing gg is asymptotically linear as |u|→∞|u|→∞ and symmetric in uu, we obtain infinitely many geometrically distinct solutions. We also consider the situation where gg is super linear with mild assumptions different from those studied previously, and establish the existence and multiplicity.