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

Abstract

We study the asymptotic behaviour, as $p\to 1^+$, of the solutions of the following inhomogeneous Robin boundary value problem: P\begin{equation*} \begin{cases} \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{cases} \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 $\lambda \ge 0$. 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 (P).