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

*(English)*Zbl 1090.35077Based on new information concerning strongly indefinite functionals without Palais-Smale conditions, we study existence and multiplicity of solutions of the Schrödinger equation

$$-{\Delta}u+V\left(x\right)u=g(x,u)\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}x\in {\mathbb{R}}^{N},\phantom{\rule{1.em}{0ex}}u\left(x\right)\to 0\phantom{\rule{1.em}{0ex}}\text{as}\phantom{\rule{4.pt}{0ex}}\left|x\right|\to \infty ,$$

where $V$ and $g$ are periodic with respect to $x$ and 0 lies in a gap of $\sigma (-{\Delta}+V)$. Supposing $g$ is asymptotically linear as $\left|u\right|\to \infty $ 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.