Behaviour of solutions to $p$-Laplacian with Robin boundary conditions as $p$ goes to $1$

TL;DR

This paper investigates the asymptotic behavior of solutions to a p-Laplacian boundary value problem with Robin conditions as p approaches 1, identifying thresholds for solution blow-up or decay and exploring the limiting 1-Laplacian case.

Contribution

It provides a detailed analysis of the solution behavior as p approaches 1, including thresholds for blow-up and decay, and studies the formal limit leading to the 1-Laplacian problem.

Findings

Identified the threshold for solution decay to zero.

Determined the threshold for solution blow-up.

Explored the formal limit as p approaches 1, leading to the 1-Laplacian.

Abstract

We study the asymptotic behaviour, as $p\to 1^{+}$, of the solutions of the following inhomogeneous Robin boundary value problem: \begin{equation} \label{pbabstract} \tag{P} \left\{\begin{array}{ll} \displaystyle -\Delta_p u_p = f & \text{in }\Omega, \displaystyle |\nabla u_p|^{p-2}\nabla u_p\cdot \nu +\lambda |u_p|^{p-2}u_p = g& \text{on } \partial\Omega, \end{array}\right. \end{equation} where $\Omega$ is a bounded domain in $\mathbb R^{N}$ with sufficiently smooth boundary, $\nu$ is its unit outward normal vector and $\Delta_p v$ is the $p$-Laplacian operator with $p>1$. The data $f\in L^{N,\infty}(\Omega)$ (which denotes the Marcinkiewicz space) and $\lambda,g$ are bounded functions defined on $\partial\Omega$ with $\lambda\ge0$. We find the threshold below which the family of $p$--solutions goes to 0 and above which this family blows up. As a second interest we deal with the $1$-Laplacian problem formally arising by taking $p\to 1^+$ in \eqref{pbabstract}.