Existence of solutions for some quasilinear elliptic system with weight and measure-valued right hand side

Let \(\Omega \) be an open bounded domain in \(I\!\!R^{n},\) we prove the existence of a solution u for the nonlinear elliptic system

$$\begin{aligned} \text{(QES) } \left\{ \begin{array}{ll} -div\sigma \left( x,u\left( x\right) ,Du\left( x\right) \right) = \mu &{}\quad \text{ in } \Omega \\ u = 0 &{}\quad \text{ on } \partial \Omega , \end{array} \right. \end{aligned}$$

(0.1)

where \(\mu \) is Radon measure on \(\Omega \) with finite mass. In particular, we show that if the coercivity rate of \(\sigma \) lies in the range \(]\frac{s+1}{s},(\frac{s+1}{s})(2-\frac{1}{n})]\) with \(s\in \left( \frac{n}{p}\,\ \infty \right) \cap \left( \frac{1}{p-1}\,\ \infty \right) ,\) then u is approximately differentiable and the equation holds with Du replaced by \(\text{ apDu }\). The proof relies on an approximation of \(\mu \) by smooth functions \(f_{k}\) and a compactness result for the corresponding solutions \(u_{k}.\) This follows from a detailed analysis of the Young measure \(\{\delta _{u}(x)\otimes \vartheta (x)\}\) generated by the sequence \({(u_{k},Du_{k})}\), and the div-curl type inequality \(\langle \vartheta (x),\sigma (x,u,\cdot )\rangle \le \overline{\sigma }(x)\langle \vartheta (x),\cdot \rangle \) for the weak limit \(\overline{\sigma }\) of the sequence.