In our work, we establish the existence of standing waves to a nonlinear Schrödinger equation with inverse-square potential on the half-line. We apply a profile decomposition argument to overcome the difficulty arising from the non-compactness of the setting. We obtain convergent minimizing sequences by comparing the problem to the problem at “infinity” (i.e., the equation without inverse square potential). Finally, we establish orbital stability/instability of the standing wave solution for mass subcritical and supercritical nonlinearities respectively.

中文翻译：

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驻波在半线上具有平方反比的非线性薛定谔方程的存在性和轨道稳定性

在我们的工作中，我们将驻波的存在性建立为在半线上具有平方反比势的非线性薛定谔方程。我们应用配置文件分解参数来克服由于设置的非紧凑性而引起的困难。我们通过将问题与“无穷大”处的问题（即没有平方反比势的方程）进行比较来获得收敛最小化序列。最后，我们分别建立了质量亚临界和超临界非线性驻波解的轨道稳定性/不稳定性。