Ellipticity conditions for the extended MHD Grad-Shafranov-Bernoulli equilibrium equations

In this study we find the points of transition between elliptic and hyperbolic regimes for the axisymmetric extended MHD equilibrium equations. The ellipticity condition is more involved than the corresponding two-fluid ones due to the imposition of the quasineutrality condition, and more complicated than the Hall MHD ellipticity condition. In fact, the inclusion of electron inertia is responsible for peculiar results; namely, even the static equilibrium equations can become hyperbolic.