Rigidity Results for the p-Laplacian Poisson Problem with Robin Boundary Conditions

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