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

TL;DR

This paper investigates a nonlinear ODE inspired by liquid crystal models, establishing existence, uniqueness, and qualitative properties of solutions crucial for understanding their stability.

Contribution

It provides new existence and uniqueness results for solutions of a nonlinear ODE related to liquid crystal models, including sign-changing solutions and stability properties.

Findings

Existence of positive solutions under general conditions

Uniqueness of solutions for a class of nonlinearities

Qualitative properties relevant to energetic stability

Abstract

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