Semi-classical states for logarithmic Schrödinger equations

In this paper, we investigate semi-classical bound states for logarithmic Schrödinger 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.