Infinitely many positive solutions to some scalar field equations with non-symmetric coefficients

In this paper the equation $ -\Delta u+a(x)u=|u|^{p-1}u \mbox{ in }\R^N$ is considered, when $N \ge2$, $p>1,\ p<{\frac{N+2}{N-2}},$ if $N\ge 3.$ Assuming that the potential $a(x)$ is a positive function belonging to $L^{N/2}_ {loc}(\R^N),$ such that $a(x)\to a_\infty > 0, \ \mbox{as} \ |x|\rightarrow \infty$, and that satisfies slow decay assumptions, but not requiring any symmetry property, the existence of infinitely many positive solutions, by purely variational methods, is proved. The shape of the solutions is described and, furthermore, their asymptotic behavior when $|a(x) - a_\infty|_ {L^ {N/2}_ {loc}(\R^N)} \to 0$.