Steady states of gas ionization with secondary emission

We consider the steady states of a gas between two parallel plates that is ionized by a strong electric field so as to create a plasma. There can be a cascade of electrons due both to the electrons colliding with the gas molecules and to the ions colliding with the cathode (secondary emission). We use global bifurcation theory to prove that there is a one-parameter family $\mathscr{K}$ of such steady states with the following property. The curve $\mathscr{K}$ begins at the sparking voltage and either the particle density becomes unbounded or $\mathscr{K}$ ends at an anti-sparking voltage. These critical voltages are characterized explicitly.