We are concerned with the existence and multiplicity of homoclinic solutions for the second order Hamiltonian equation
$$-\ddot{u}+\omega(t)u=F_u(t,u) \quad t \in \mathbb{R}, \quad\quad\quad(1)$$
where
$${\omega \in \mathcal{C}(\mathbb{R})}$$
is positive and bounded, and
$${F\in \mathcal{C}^1(S^1\times\mathbb{R})}$$
. Under some growth condition on *F*, we prove that (1) admits at least two solutions which are homoclinic to zero and do not change sign. We also prove that for every integer *k* ≥ 1, (1) possesses at least two solutions homoclinic to zero changing sign exactly *k* times, and for *k* ≥ 2 these solutions have at least *k* and at most *k* + 2 zeros which are isolated, or ‘isolated from the left’, or ‘isolated from the from right’.