Normalized solutions to mass supercritical Schr\"odinger equations with radial potentials

We study the stationary nonlinear Schr\"odinger equation \begin{equation}-\Delta u+V(x)u+\lambda u=|u|^{q-2}u,\quad u \in H^1(\mathbb{R}^N), \quad N \geq 2\end{equation} where $V \in L^{\infty}(\mathbb{R}^N)$ is a radial potential. In the $L^2$-supercritical regime, we show the existence of an explicit $\mu_0 >0$ such that, for any $\mu \in (0, \mu_0)$, the equation admits two solutions having $L^2$ norm $\mu$. The potential $V$ is not assumed to have a sign, nor a specific behavior at infinity and only a low regularity is required. Our proof relies on the use of Morse type information, on some spectral arguments, and on a blow-up analysis developed in a radial setting.