$p$-Laplace Serrin problem on planar annuli

We prove a rigidity result for the problem $Δ_p u=-2$ defined on a ring-shaped domain in $\mathbb{R}^2$, when homogeneous Dirichlet boundary conditions are imposed and $p\in(1,2)$. This problem admits a natural physical interpretation, as it describes the steady laminar flow of a generalised Newtoni