The paper deals with the one-dimensional isothermal steady hydrodynamic model (i.e., the current density and lattice temperature are assumed equal to \(1\) in the Euler-Poisson equations) \[ \begin{cases} \left( 1- \dfrac{1}{n^{2}} \right) n_{x} = nE - \dfrac{1}{\tau}, \\ \, E_{x} = n - b(x), \quad x\in(0,1) \end{cases} \] with sonic boundary conditions \(n(0) = n(1) = 1\). In the above system, \(n=n(x)\) denotes the electron density, \(E=E(x)\) the electric field, \(b=b(x)\) the doping profile, \(\tau>0\) the momentum relaxation time. The authors provide the existence, uniqueness and stability for solutions \((n,E)\) in the case of an approximate problem depending on a real parameter. The results are also confirmed by numerical simulations.