Normalized solutions of mass supercritical Schrödinger equations with potential

This paper is concerned with the existence of normalized solutions of the nonlinear Schrodinger equation-Delta u + V(x)u + lambda u = vertical bar u vertical bar(p-2)u in R-Nin the mass supercritical and Sobolev subcritical case 2 + 4/N < p < 2*. We prove the existence of a solution (u, lambda) is an element of H-1 (R-N) x R+ with prescribed L-2-norm parallel to u parallel to(2) = rho under various conditions on the potential V : R-N -> R, positive and vanishing at infinity, including potentials with singularities. The proof is based on a new min-max argument.