Existence of a positive solution to a nonlinear scalar field equation with zero mass at infinity

We establish the existence of a positive solution to the problem $$-\mathrm{\Delta }u+V(x)u=f(u),u\in D^{1,2}({\mathbb{R}}^N),$$ for $N\ge 3$, when the nonlinearity $f$ is subcritical at infinity and supercritical near the origin, and the potential $V$ vanishes at infinity. Our result includes situations in which the problem does not have a ground state. Then, under a suitable decay assumption on the potential, we show that the problem has a positive bound state.