Multiple Solutions for Scalar Field Equations with Potentials having "Subsidences"

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.