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Homoclinic orbits for Schrödinger systems. (English) Zbl 1195.35281

The authors consider a Schrödinger equation with an arbitrary nonlinear term and a potential periodic in the space and time variables. They assume that the Schrödinger equation has a stationary solution at zero and prove the existence of a nonstationary solution homoclinic to the zero stationary solution in the space and time.The authors analyze a weak superlinear case and an asymptotically linear case without the Ambrosetti-Rabinowitz global superquadratic condition. The main goal is to prove boundedness of a Palais-Smale sequence using directly a weak linking theorem for a modified functional and thereby have a sequence of critical points, which provides a nontrivial solution of the Schrödinger equation.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
35K55 Nonlinear parabolic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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