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An asymptotically periodic Schrödinger equation with indefinite linear part. (English) Zbl 1056.35065

Summary: We consider the Schrödinger equation \(-\Delta u+ V(x)u= f(x,u)\), where \(V\) is periodic and \(f\) asymptotically periodic in the \(x\)-variables, \(0\) is in a spectral gap of \(-\Delta+ V\) and \(f\) is either asymptotically linear or superlinear as \(|u|\to\infty\). We show that this equation has a solution \(u\in H^1(\mathbb{R}^N)\), \(u\neq 0\).

MSC:

35J60 Nonlinear elliptic equations
35B15 Almost and pseudo-almost periodic solutions to PDEs
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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