Renormalized Solutions for Quasilinear Elliptic Equations with Robin Boundary Conditions, Lower-Order Terms, and $L^1$ Data

Abstract

In this paper, we establish the existence of a solution for a class of quasilinear equations characterized by the prototype: \begin{equation} \left\{\begin{aligned} -\operatorname{div}(\vartheta_\alpha|\nabla u|^{p-2} \nabla u)+\vartheta_\gamma b|\nabla u|^{p-1}+\vartheta_\gamma c|u|^{r-1} u & =f \vartheta_\alpha & & \text { in } \Omega, \\ \vartheta_\alpha|\nabla u|^{p-2} \nabla u \cdot \nu+\vartheta_\beta|u|^{p-2} u & =g \vartheta_\beta & & \text { on } \partial \Omega . \end{aligned}\right. \end{equation} Here, $\Omega$ is an open subset of $\mathbb{R}^N$ with a Lipschitz boundary, where $N\geq 2$ and $1 < p < N$. We define $\vartheta_a(x) = (1 + |x|)^a$ for $a \in (-N, (p-1)N)$, and the constants $\alpha, \beta, \gamma, r$ satisfy suitable conditions. Additionally, $f$ and $g$ are measurable functions, while $b$ and $c$ belong to a Lorentz space. Our approach also allows us to establish stability results for renormalized solutions.