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

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_α|\nabla u|^{p-2} \nabla u)+\vartheta_γb|\nabla u|^{p-1}+\vartheta_γc|u|^{r-1} u & =f \vartheta_α& & \text { in } Ω, \ \vartheta_α|\nabla u|^{p-2} \nabla u \cdot ν+\vartheta_β|u|^{p-2} u & =g \vartheta_β& & \text { on } \partial Ω. \end{aligned}\right. \end{equation} Here, $\mathrm{\Omega }$ is an open subset of ${\mathbb{R}}^N$ with a Lipschitz boundary, where $N\ge 2$ and $1<p<N$. We define ${\vartheta }_a(x)=(1+\mathrm{\mid }x\mathrm{\mid }{)}^a$ 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.