Normalized solutions to mass supercritical Schrodinger equations with negative potential

TL;DR

This paper investigates the existence and nonexistence of positive normalized solutions to a mass supercritical Schrödinger equation with a nonnegative potential, providing conditions for multiple solutions and ruling out others.

Contribution

It establishes new existence results for solutions under explicit smallness and mass constraints, including mountain pass and local minimizer solutions, and proves nonexistence in certain regimes.

Findings

Existence of mountain pass solutions at positive energy under small potential

Existence of a negative energy local minimizer when mass is sufficiently small

Nonexistence of solutions with negative energy under certain conditions

Abstract

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