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Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms. (English) Zbl 1090.35077

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 N ,u(x)0as|x|,

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 superlinear with mild assumptions different from those studied previously, and establish the existence and multiplicity.

35J60Nonlinear elliptic equations
35Q55NLS-like (nonlinear Schrödinger) equations