## 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 $-\Delta u+V(x)u=g(x,u)\quad\text{for }x\in\mathbb R^N,\quad u(x)\to 0\quad \text{as }|x|\to\infty,$ 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 $$|u|\to \infty$$ 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.

### MSC:

 35J60 Nonlinear elliptic equations 35Q55 NLS equations (nonlinear Schrödinger equations)
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### References:

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