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

Abstract:

In this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems whose prototype is \begin{equation}\tag {P} \begin {cases} - \bigtriangleup _p u -\operatorname {div}(c(x)|u|^{\gamma })+b(x)|\nabla u|^{\lambda } =\mu & \text {in $\Omega $},\\ u=0 & \text {in $\partial \Omega $}, \end{cases} \end{equation} 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 $|c|$ 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 $\gamma =\lambda =p-1$, $|c|$ belongs to the Lorentz space $L^{\frac {N}{p-1},r}(\Omega )$, $\frac {N}{p-1}\leq r<+\infty$, and $\|c\|_{ L^{\frac {N}{p-1},r}(\Omega )}$ is small enough.