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Existence of nontrivial solutions for asymptotically linear periodic Schrödinger equations. (English) Zbl 1312.35056

Summary: We study the Schrödinger equation: \[ -\Delta u + V (x)u = f(x, u), \quad u \in H^1(\mathbb{R}^N), \] where \(V\) is periodic and \(f\) is periodic in the \(x\)-variables, 0 is in a gap of the spectrum of the operator \(-\Delta + V\) and \(f\) is asymptotically linear as \(|u| \rightarrow +\infty.\) We prove that under some asymptotically linear assumptions for \(f\), this equation has a nontrivial solution. Our assumptions for \(f\) are different from the classical assumptions raised by G. Li and A. Szulkin [Commun. Contemp. Math. 4, No. 4, 763–776 (2002; Zbl 1056.35065)].

MSC:

35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations

Citations:

Zbl 1056.35065
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References:

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