Normalized solutions of mass supercritical Schrödinger equations with potential

This paper is concerned with the existence of normalized solutions of the nonlinear Schr"odinger equation [ -Delta u+V(x)u+lambda u = |u|^{p-2}u qquad ext{in $mathbb{R}^N$} ] in the mass supercritical and Sobolev subcritical case $2+rac{4}{N}<2^*$. We prove the existence of a solution $(u,lambda)in H^1(mathbb{R}^N) imesmathbb{R}^+$ with prescribed $L^2$-norm $|u|_2= ho$ under various conditions on the potential $V:mathbb{R}^N omathbb{R}$, positive and vanishing at infinity, including potentials with singularities. The proof is based on a new min-max argument.