Non-uniqueness of transonic shock solutions to Euler–Poisson system with varying background charges

The Euler–Poisson equations with varying background charges in finitely long flat nozzles are investigated, for which two and only two transonic shock solutions are constructed. In href{https://dx.doi.org/10.4310/CMS.2012.v10.n2.a1}{[\textrm{T. Luo and Z.P. Xin, Commun. Math. Sci., 10:419–462, 2012}]}, Luo and Xin established the wellposedness of steady Euler–Poisson equations for the constant background charge. Motivated by their pioneering work and combined with the special physical character of semiconductor devices, we propose the transonic shock problem in which the density of the background charge is a piecewise constant function and its discontinuity is determined only by shock fronts. The existence and non-uniqueness of transonic shock solutions are obtained via the method of shock matching.