## Existence of nontrivial solutions for periodic Schrödinger equations with new nonlinearities.(English)Zbl 1474.35295

Summary: We study the Schrödinger equation: $$- \Delta u + V \left(x\right) u + f \left(x, u\right) = 0, u \in H^1(\mathbb{R}^N)$$, where $$V$$ is $$1$$-periodic and $$f$$ is $$1$$-periodic in the $$x$$-variables; $$0$$ is in a gap of the spectrum of the operator $$- \Delta + V$$. We prove that, under some new assumptions for $$f$$, this equation has a nontrivial solution. Our assumptions for the nonlinearity $$f$$ are very weak and greatly different from the known assumptions in the literature.

### MSC:

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

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