Stability of point defects of degree $\pm \frac{1}{2}$ in a two-dimensional nematic liquid crystal model

We study $k$-radially symmetric solutions corresponding to topological defects of charge $\frac{k}{2}$ for integer $k\mathrm{\ne }0$ in the Landau-de Gennes model describing liquid crystals in two-dimensional domains. We show that the solutions whose radial profiles satisfy a natural sign invariance are stable when $\mathrm{\mid }k\mathrm{\mid }=1$ (unlike the case $\mathrm{\mid }k\mathrm{\mid }>1$ which we treated before). The proof crucially uses the monotonicity of the suitable components, obtained by making use of the cooperative character of the system. A uniqueness result for the radial profiles is also established.