## On a semilinear Schrödinger equation with periodic potential.(English)Zbl 0952.35047

The existence of a nontrivial solution for the Schrödinger equation $$-\Delta u+V(x)u=f(x,u)$$, $$x\in \mathbb{R}^N$$ is studied for indefinite periodic potentials $$V$$ and superlinear, subcritical nonlinearities $$f$$ in $$u$$ and with the same period of $$f$$ in $$x$$ as $$V.$$ Under the assumption that $$0$$ is in a gap of the spectrum of $$-\Delta+V$$ it is proved the existence of nontrivial solutions having finite energy. The proof is based on the approximation of the considered problem by periodic problems in cubes.

### MSC:

 35J60 Nonlinear elliptic equations 35B10 Periodic solutions to PDEs 35A35 Theoretical approximation in context of PDEs
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### References:

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