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

Abstract

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.