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One-dimensional scalar field equations involving an oscillatory nonlinear term

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


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