Semilinear elliptic problems in $\mathbb{R}^N$: the interplay between the potential and the nonlinear term

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.