Normalized solutions to mass supercritical Schrodinger equations with negative potential

Abstract

We study the existence of positive solutions with prescribed $L^2$-norm for the Schrödinger equation \[ -Δu-V(x)u+λu=|u|^{p-2}u\qquadλ\in \mathbb{R},\quad u\in H^1(\mathbb{R}^N), \] where $V\ge 0$, $N\ge 1$ and $p\in \left(2+\frac{4}{N},2^{\ast }\right)$, $2^{\ast }:=\frac{2N}{N-2}$ if $N\ge 3$ and $2^{\ast }:=+\infty$ 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.

In this version we have a little changed the title of the paper