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_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, $Ω$ 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 $α, β, γ, r$ 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.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Remark 2.2
• Proposition 2.3
• Lemma 2.4
• Proposition 2.5
• Proposition 2.6
• Definition 2.7
• Remark 2.8