Summary: In this paper, we analyze the existence and the numerical simulation of a capacity solution to a coupled nonlinear elliptic system, whose unknowns are the temperature inside a semiconductor material \(u\), and the electric potential \(\varphi\). The model problem we refer to is \[ \begin{cases} -\Delta_{\vec{p}}\, u =\rho(u)|\nabla \varphi|^2 & \text{ in } \Omega \\ \operatorname{div}(\rho(u) \nabla \varphi)=0 & \text{ in } \Omega \\ \varphi=\varphi_0 & \text{ on } \partial \Omega \\ u=0 & \text{ on } \partial \Omega \end{cases} \] where \(\Omega\) is an open bounded set of \(\mathbb{R}^N\), \(N\geq 2\) and \(\Delta_{\vec{p}}\, u= \sum \limits_{i = 1}\limits^N \partial_i\left( |\partial_i u|^{p_i-2}\partial_i u \right)\), is the \(\vec{p}\)-Laplacian operator. We consider the case of a nonuniformly elliptic problem.