Rigidity results for the $p$-Laplacian Poisson problem with Robin boundary conditions

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be an open, bounded and Lipschitz set. We consider the Poisson problem for the $p-$Laplace operator associated to $\mathrm{\Omega }$ with Robin boundary conditions. In this setting, we study the equality case in the Talenti-type comparison stated in \cite{AGM}. We prove that the equality is achieved only if $\mathrm{\Omega }$ is a ball and both the function $u$ and the right hand side $f$ of the Poisson equation are radial.