In this paper, the authors study the following class of quasilinear elliptic equations \[ \begin{cases} -\mathrm{div}(A(x, u)\nabla u)+u=\mu,\quad &\text{in}\,\, \Omega,\\ u=0 \quad &\text{on}\,\, \partial\Omega,\end{cases}\tag{1} \] where \(\Omega\) is a bounded domain of \(\mathbb R^N,\; N\geq 3\) and \(\mu \in {M}(\Omega)\) is a finite Radon measure. \(A(x, s): \Omega \times \mathbb R \rightarrow \mathbb R^{N^2}\) is a bounded symmetric Carathéodory matrix function satisfying \[ \frac{\alpha |\xi|^2}{(1+|s|)^\gamma}\leq \langle A(x, s)\xi, \xi \rangle \leq \frac{\beta |\xi|^2}{(1+|s|)^\gamma} \] for all \(\xi \in \mathbb R^N, s\in \mathbb R\) and a.e. \(x\in \Omega,\) where \(\alpha, \beta\) and \(\gamma\) are positive constants. The authors prove the existence result to problem \((1)\) when \(0<\gamma<1\), regularity properties in the case \(0<\gamma<\frac{2}{N}\) and qualitative properties of solutions for \(\frac{2}{N}\leq\gamma<1\). The existence and uniqueness of weak entropy solutions are also established.