Multiple nodal solutions to a scalar field equation with double-power nonlinearity and zero mass at infinity

We consider the nonlinear elliptic equation \begin{equation*} -Δu + V(x)u = f(u), \qquad u\in D^{1,2}_0(Ω), \end{equation*} in an exterior domain $\mathrm{\Omega }$ of ${\mathbb{R}}^N$, where $V$ is a scalar potential that decays to zero at infinity and the nonlinearity $f$ is subcritical at infinity and supercritical near the origin. Under weak symmetry assumptions, we provide conditions that guarantee that this problem has a prescribed number of sign-changing solutions. In particular, we show that in dimensions $N\ge 4$ there are numerous examples of exterior domains with finite symmetries in which the problem has a predetermined number of nodal solutions.