On the stability of radial solutions to an anisotropic Ginzburg-Landau equation

We study the linear stability of entire radial solutions $u(re^{i\theta})=f(r)e^{i\theta}$, with positive increasing profile $f(r)$, to the anisotropic Ginzburg-Landau equation \[ -\Delta u -\delta (\partial_x+i\partial_y)^2\bar u =(1-|u|^2)u,\quad -1<\delta <1, \] which arises in various liquid crystal models. In the isotropic case $\delta=0$, Mironescu showed that such solution is nondegenerately stable. We prove stability of this radial solution in the range $\delta\in (\delta_1,0]$ for some $-1<\delta_1<0$, and instability outside this range. In strong contrast with the isotropic case, stability with respect to higher Fourier modes is \emph{not} a direct consequence of stability with respect to lower Fourier modes. In particular, in the case where $\delta\approx -1$, lower modes are stable and yet higher modes are unstable.