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.

### One dimensional scalar field equations involving an oscillatory nonlinear term

#### 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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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11769/8395
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