One-dimensional scalar field equations involving an oscillatory nonlinear term

Francesca Faraci; Alexandru Kristály

doi:10.3934/dcds.2007.18.107

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2007, Volume 18, Issue 1: 107-120. Doi: 10.3934/dcds.2007.18.107

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One-dimensional scalar field equations involving an oscillatory nonlinear term

Francesca Faraci^1, and
Alexandru Kristály^2,

1.
University of Catania, Department of Mathematics, 95125 Catania
2.
Babeş-Bolyai University, Department of Economics, 400591 Cluj-Napoca

Received: May 31, 2006

Revised: November 30, 2006

Published: December 2006

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Abstract

In this paper we study the equation $-u''+V(x)u=W(x)f(u),\ x\in\mathbb{R},$ where the nonlinear term $f$ has certain oscillatory behaviour. Via two different variational arguments, we show the existence of infinitely many homoclinic solutions whose norms in an appropriate functional space which involves the potential $V$ tend to zero (resp. at infinity) whenever $f$ oscillates at zero (resp. at infinity). Unlike in classical results, neither symmetry property on $f$ nor periodicity on the potentials $V$ and $W$ are required.

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Mathematics Subject Classification: Primary: 35J60; Secondary: 35J65.

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