  It is considered a semilinear elliptic partial differential equation in
$\mathbb{R}^N$ with a potential that may vanish at infinity and a nonlinear
term with subcritical growth. A positive solution is proved to exist depending
on the interplay between the decay of the potential at infinity and the
behavior of the nonlinear term at the origin. The proof is based on a
penalization argument, variational methods, and $L^\infty$ estimates. Those
estimates allow dealing with settings where the nonlinear source may have
supercritical, critical, or subcritical behavior near the origin. Results that
provide the existence of multiple and infinitely many solutions when the
nonlinear term is odd are also established.
