On the \(N\)-dimensional stationary drift-diffusion semiconductor equations (Q1588349)

The paper under review is concerned with the following system of PDE's in a bounded domain \(\Omega \subset \mathbb{R}^{N}\): \[ \begin{aligned} -&\operatorname {div} (\sigma (u)\nabla \psi)= p-n+f ,\tag{1}\\ -&\operatorname {div} J_{n}= r(n,p)(1-np)+g, \tag{2}\\ &\operatorname {div} J_{p}= r(n,p)(1-np)+g, \tag{3}\\ -&\operatorname {div}(k(u)\nabla u)= \operatorname {div} (a(\psi)J_{n}+b(\psi)J_{p}), \tag{4} \end{aligned} \] where \[ J_{n}=D_{1}\nabla n - \mu_{1}n\sigma (u)\nabla \psi , \quad J_{p}= -D_{2}\nabla p - \mu_{2} p\sigma (u)\nabla \psi \] \parindent 0cm are the current density of the concentrations. Here the unknown functions \(\psi , n, p\), and \(T\) represent the electrostatic potential, the electron density, the hole density, and the temperature, respectively. These functions are subject to mixed boundary conditions on \(\partial \Omega\). Under physically natural conditions on the functions \(\sigma ,K, r\) as well as \(a,b\) and on the other known quantities, the authors prove the existence of at least one weak solution to (1)-(4) satisfying \[ \|\psi \|_{L^{\infty}(\Omega)}+ \|n\|_{L^{\infty}(\Omega)}+ \|p\|_{L^{\infty}(\Omega)}< + \infty . \] The method of proof consists in solving a modified system in place of (1)-(4), establishing a priori estimates on the solutions to the modified system and then carrying out the passage to limit to obtain the existence of a weak solution to the system under consideration.