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

Let $Ω\subset \mathbb{R}^n$ be an open, bounded and Lipschitz set. We consider the Poisson problem for the $p-$Laplace operator associated to $Ω$ 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 $Ω$ is a ball and both the function $u$ and the right hand side $f$ of the Poisson equation are radial.

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