A Singular Boundary-value Problem Arising in a Pre-breakdown Gas-discharge

The nonlinear two-point boundary value problem epsilon xy double prime plus (g(x)-y)y prime equals O,y(O) equals O,y(R) equals k, where g is a given function. It is proved that the problem has a unique solution, and the limiting behavior of this solution is studied as R approaches infinity and as epsilon approaches zero from above. Furthermore, it is shown how a so-called pre-breakdown discharge in an ionized gas between two electrodes can be described by an equation of this form, and we interpret the results physically.