Capacity Solution to a Nonlinear Elliptic Coupled System in Orlicz–Sobolev Spaces

Abstract

We analyze the existence of a capacity solution to the following nonlinear elliptic coupled system, whose unknowns are the temperature inside a semiconductor material, u, and the electric potential, $\varphi$,

$$\begin{aligned} \left\{ \begin{array}{ll} -Au= \rho (u)|\nabla \varphi |^2 &{}\quad \mathrm{in}\ \Omega , \\ \mathop {\mathrm{div}}\nolimits (\rho (u)\nabla \varphi ) =0&{}\quad \mathrm{in}\ \Omega , \\ \varphi =\varphi _0 &{}\quad \hbox {on } \partial \Omega , \\ u=0&{}\quad \mathrm{on}\, {\partial \Omega }, \end{array} \right. \end{aligned}$$

where $\Omega \subset {\mathbb {R}}^d$, $d\ge 2$ and $\displaystyle Au=-\mathop {\mathrm{div}}\nolimits a(x,u,\nabla u)$ is a Leray–Lions operator defined on $W_0^{1}L_M(\Omega )$, M is a N-function which does not have to satisfy a $\Delta _2$ condition. Therefore, we work with generalized Orlicz–Sobolev spaces which are not necessarily reflexive. The function $\varphi _0$ is given. The proof combines truncation methods, monotonicity techniques and regularizing methods in Orlicz spaces. We introduce a sequence of approximate problems which converges (up to a subsequence) in a certain sense to a capacity solution in the context of non-reflexive Orlicz–Sobolev spaces.