We consider the problem of uniqueness of ground states of prescribed mass for the Nonlinear Schrodinger Energy with power nonlinearity on noncompact metric graphs. We first establish that the Lagrange multiplier appearing in the NLS equation is constant on the set of ground states of mass $\mu$, up to an at most countable set of masses. Then we apply this result to obtain uniqueness of ground states on two specific noncompact graphs. Finally we construct a graph that admits at least two ground states with the same mass having different Lagrange multipliers. Our proofs are based on careful variational arguments and rearrangement techniques, and hold both for the subcritical range $p\in(2,6)$ and in the critical case $p = 6$.

中文翻译：

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度量图上规定质量 NLS 基态的唯一性和非唯一性

我们在非紧度量度图上考虑具有功率非线性的非线性薛定谔能量的规定质量基态的唯一性问题。我们首先确定出现在 NLS 方程中的拉格朗日乘数在质量为 $\mu$ 的基态集合上是常数，直到最多可数的质量集合。然后我们应用这个结果来获得两个特定非紧图上基态的唯一性。最后，我们构建了一个图，该图允许至少两个具有相同质量且具有不同拉格朗日乘子的基态。我们的证明基于仔细的变分论证和重排技术，并且在亚临界范围 $p\in(2,6)$ 和临界情况下 $p = 6$ 都成立。