Abstract

\bfA \bfb \bfs \bft \bfr \bfa \bfc \bft . We study the linear stability of entire radial solutions u(rei\theta ) = f(r)ei\theta , with positive increasing profile f(r), to the anisotropic Ginzburg-Landau equation - \Delta u - \delta (\partial x+i\partial y)2u\= = (1 - | u| 2)u, - 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 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.