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We study a singular nonlinear ordinary differential equation on intervals$[0,R)$ with $R\le +\infty$, motivated by the Ginzburg-Landau models insuperconductivity and Landau-de Gennes models in liquid crystals. We proveexistence and uniqueness of positive solutions under general assumptions on thenonlinearity. Further uniqueness results for sign-changing solutions areobtained for a physically relevant class of nonlinearities. Moreover, we provea number of fine qualitative properties of the solution that are important forthe study of energetic stability.

Uniqueness results for an ODE related to a generalized Ginzburg-Landau model for liquid crystals