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.


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