The energy super-critical Gross–Pitaevskii equation with a harmonic potential is revisited in the particular case of cubic focusing nonlinearity and dimension $d\ge 5$. In order to prove the existence of a ground state (a positive, radially symmetric solution in the energy space), we develop the shooting method and deal with a one-parameter family of classical solutions to an initial-value problem for the stationary equation. We prove that the solution curve (the graph of the eigenvalue parameter versus the supremum norm) is oscillatory for $d\le 12$ and monotone for $d\ge 13$. Compared to the existing literature, rigorous asymptotics are derived by constructing three families of solutions to the stationary equation with functional-analytic rather than geometric methods.

在三次聚焦非线性和维数的特殊情况下，重新讨论了具有谐波电位的能量超临界Gross-Pitaevskii方程 $d\ge 5$。为了证明存在基态（能量空间中的一个正，径向对称解），我们开发了一种射击方法，并针对该平稳方程的初值问题，处理了一类经典的单参数解决方案。我们证明了解曲线（特征值参数与最高范数的关系图）对于$d\le 12$ 和单调 $d\ge 13$。与现有文献相比，通过使用函数分析而非几何方法构造固定方程的三个解系列，可以得出严格的渐近性。