Normalized solutions to mass supercritical Schrödinger equations with negative potential

We study the existence of positive solutions with prescribed L2-norm for the mass supercritical Schrödinger equation −Δu+λu−V(x)u=|u|^(p−2)uu∈H^1(R^N),λ∈R, where V≥0, N≥1 and [Formula presented], [Formula presented] if N≥3 and 2⁎:=+∞ if N=1,2. We treat two cases. Firstly, under an explicit smallness assumption on V and no condition on the mass, we prove the existence of a mountain pass solution at positive energy level, and we exclude the existence of solutions with negative energy. Secondly, requiring that the mass is smaller than some explicit bound, depending on V, and that V is not too small in a suitable sense, we find two solutions: a local minimizer with negative energy, and a mountain pass solution with positive energy. Moreover, a nonexistence result is proved.