On the stationary semiconductor equations arising in modeling an LBIC technique (Q1911773)

The paper deals with the system \[ \begin{aligned} - \nabla & \cdot (\varepsilon \nabla u)+ e^{u+ v}- e^{- u+ w}- N= 0,\tag{1}\\ - \nabla & \cdot (\mu_1 e^{u+ v} \nabla v)+ Q(e^{v+ w}- 1)- g= 0,\tag{2}\\ - \nabla & \cdot (\mu_2 e^{- u+ w} \nabla w)+ Q(e^{v+ w}- 1)- g= 0,\tag{3}\end{aligned} \] in \(\Omega\subset \mathbb{R}^d\) with boundary conditions (4) \(u= \overline u\), \(v= 0\), \(w= 0\) on \(\Sigma_D\), and (5) \(\partial u/\partial \nu= \partial v/ \partial\nu= \partial w/ \partial\nu= 0\) on \(\Sigma_N\), \(\Sigma_D \cup \Sigma_N= \partial \Omega\). The authors prove that if \(g\in L^2(\Omega)\) with \(C|g|_2\leq 1/2\), then there is at least one weak solution \((u, v, w)\) of (1)--(5) such that (6) \(|u- u_0|_\infty\), \(|v|_\infty\), \(|w|_\infty\leq C|g|_2\), where \(u_0\) is the equilibrium potential, and if \(|g|_2\) is sufficiently small, then there is at most one weak solution of (1)--(5) which satisfies the estimates (6). The existence result is obtained also in the case of constant mobilities and permitivity (\(\mu_1\), \(\mu_2\) and \(\varepsilon\) are constants) without any restriction on the size of \(g\).