Multiple solutions of Schrodinger 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 Schrodinger equation -Delta u + V(x)u = g(x,u) for x is an element of R-N, u(x) -> 0 as vertical bar x vertical bar -> infinity. where V and g are periodic with respect to x and 0 lies in a gap sigma(-Delta+V). Supposing g is asymptotically linear as vertical bar u vertical bar -> infinity 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. (c) 2005 Elsevier Inc. All rights reserved.