Existence of renormalized solutions to nonlinear elliptic equations with two lower order terms and measure data

In this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems whose prototype is

$(P)$ $\left\{\begin{array}{ll} - \bigtriangleup _p u -\hbox{div }(c(x)... ... & \hbox{in} ~ \Omega, u=0 & \hbox{on} ~ \partial\Omega, \end{array}\right.$

where $\Omega$ is a bounded open subset of $\mathbb{R}^N$, $N\geq 2$, $\bigtriangleup _p$ is the so-called $p-$Laplace operator, $1< p< N$, $\mu$ is a Radon measure with bounded variation on $\Omega$, $0\le\gamma\le p-1$, $0\le\lambda\le p-1$, and $\vert c\vert$ and $b$ belong to the Lorentz spaces $L^{\frac{N}{p-1},r}(\Omega)$, $\frac{N}{p-1}\leq r \leq +\infty$, and $L^{N,1}(\Omega)$, respectively. In particular we prove the existence under the assumptions that $\gamma=\lambda=p-1$, $\vert c\vert$ belongs to the Lorentz space $L^{\frac{N}{p-1},r}(\Omega)$, $\frac{N}{p-1}\leq r<+\infty$, and $\Vert c\Vert _{ L^{\frac{N}{p-1},r}(\Omega)}$ is small enough.