  In this paper the question of finding infinitely many solutions to the
problem $-\Delta u+a(x)u=|u|^{p-2}u$, in $\mathbb{R}^N$, $u \in
H^1(\mathbb{R}^N)$, is considered when $N\geq 2$, $p \in (2, 2N/(N-2))$, and
the potential $a(x)$ is a positive function which is not required to enjoy
symmetry properties. Assuming that $a(x)$ satisfies a suitable "slow decay at
infinity" condition and, moreover, that its graph has some "dips", we prove
that the problem admits either infinitely many nodal solutions either
infinitely many constant sign solutions. The proof method is purely variational
and allows to describe the shape of the solutions.
