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: \begin{equation} \label{pbabstract} \tag{P} \left\{\begin{array}{ll} \displaystyle -Δ_p u_p = f & \text{in }Ω, \displaystyle |\nabla u_p|^{p-2}\nabla u_p\cdot ν+λ|u_p|^{p-2}u_p = g& \text{on } \partialΩ, \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 $\lambda \ge 0$. 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 \eqref{pbabstract}.