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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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