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One-dimensional scalar field equations involving an oscillatory nonlinear term
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.