We study the nonlinear Schrödinger equation for systems of N orthonormal functions. We prove the existence of ground states for all N when the exponent p of the non linearity is not too large, and for an infinite sequence \(N_j\) tending to infinity in the whole range of possible p’s, in dimensions \(d\ge 1\). This allows us to prove that translational symmetry is broken for a quantum crystal in the Kohn–Sham model with a large Dirac exchange constant.

中文翻译：

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正交函数的非线性薛定ding方程：基态的存在

我们研究N个正交函数系统的非线性Schrödinger方程。我们证明了当非线性指数p不太大时，对于所有N都存在基态，对于无限序列\（N_j \）在可能的p的整个范围内都趋于无穷大，在维\（ d \ ge 1 \）。这使我们能够证明，在具有大Dirac交换常数的Kohn-Sham模型中，量子晶体的平移对称性被打破。