We shall give the existence of a capacity solution to a nonlinear elliptic coupled system, whose unknowns are the temperature inside a semiconductor material, u, and the electric potential, \(\varphi \), the model problem we refer to is$$\begin{aligned} \left\{ \begin{array}{l} \Delta _p u+g(x,u)= \rho (u)|\nabla \varphi |^2 \quad \mathrm{in} \quad \Omega ,\\ {{\,\mathrm{div}\,}}(\rho (u)\nabla \varphi ) =0 \quad \mathrm{in} \quad \Omega ,\\ \varphi =\varphi _0 \quad \text{ on } \quad {\partial \Omega },\\ u=0 \quad \mathrm{on} \quad {\partial \Omega }, \end{array} \right. \end{aligned}$$where \(\Omega \subset \mathbb {R}^N\), \(N\ge 2\) and \(\Delta _p u=-{\text {div}}\left( |\nabla u|^{p-2} \nabla u\right) \) is the so-called p-Laplacian operator, and g a nonlinearity which satisfies the sign condition but without any restriction on its growth. This problem may be regarded as a generalization of the so-called thermistor problem, where we consider the case of the elliptic equation is non-uniformly elliptic.

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摄动非线性耦合系统的容量解

我们将给出非线性椭圆耦合系统的容量解的存在，其未知数是半导体材料内部的温度u和电势\（\ varphi \），我们提到的模型问题是$$ \ begin {aligned} \ left \ {\ begin {array} {l} \ Delta _p u + g（x，u）= \ rho（u）| \ nabla \ varphi | ^ 2 \ quad \ mathrm {in} \ quad \ Omega，\\ {{\，\ mathrm {div} \，}}（\ rho（u）\ nabla \ varphi）= 0 \ quad \ mathrm {in} \ quad \ Omega，\\ \ varphi = \ varphi _0 \ quad \ text {on} \ quad {\ partial \ Omega}，\\ u = 0 \ quad \ mathrm {on} \ quad {\ partial \ Omega}，\ end {array} \ right。\ end {aligned} $$其中\（\ Omega \ subset \ mathbb {R} ^ N \），\（N \ ge 2 \）和\（\德尔塔_p U = - {\文本{DIV}} \左（| \ nabla U | ^ {P-2} \ nabla U \右）\）是所谓的P-Laplacian算子，并且克一满足符号条件但对其增长没有任何限制的非线性。这个问题可以看作是所谓的热敏电阻问题的推广，在这里我们认为椭圆方程的情况是非均匀椭圆的。