Normalized solutions to mass supercritical Schrödinger equations with radial potentials

We study the stationary nonlinear Schrödinger equation \begin{equation}-Δu+V(x)u+λu=|u|^{q-2}u,\quad u \in H^1($\mathbb{R}$^N), \quad N \geq 2\end{equation} where $V\in L^{\mathrm{\infty }}({\mathbb{R}}^N)$ 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.