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January  2007, 18(1): 107-120. doi: 10.3934/dcds.2007.18.107

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, Italy
2. 

Babeş-Bolyai University, Department of Economics, 400591 Cluj-Napoca, Romania

Received  June 2006 Revised  December 2006 Published  February 2007

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.
Keywords: infinitely solutions., Scalar field equation, oscillatory nonlinearity.
Mathematics Subject Classification: Primary: 35J60; Secondary: 35J6.
Citation: Francesca Faraci, Alexandru Kristály. One-dimensional scalar field equations involving an oscillatory nonlinear term. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 107-120. doi: 10.3934/dcds.2007.18.107
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