In this paper, we investigate semi-classical bound states for logarithmic Schrdinger type equations with a potential function which has a finite number of singularities of at most logarithmic strength. We construct localized solutions concentrating at a logarithmic type singular point of the potential, and we also characterize the asymptotic limiting profile of the localized solutions. To accomplish these, we develop new penalization techniques for treating the difficulties associated with the non-smoothness of the variational formulation and the singularity of the potentials.

中文翻译：

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对数 Schrdinger 方程的半经典状态

在本文中，我们研究了对数 Schrdinger 型方程的半经典束缚态，其势函数具有最多对数强度的有限奇点数。我们构造了集中在势能的对数型奇异点的局部解，并且我们还表征了局部解的渐近极限曲线。为了实现这些，我们开发了新的惩罚技术来处理与变分公式的非平滑性和势能的奇异性相关的困难。