Almost rigidity of the Talenti-type comparison theorem on $\mathrm{R}\mathrm{C}\mathrm{D}(0,N)$ space

In this paper, we prove a Talenti-type comparison theorem for the $p$-Laplacian with Dirichlet boundary conditions on open subsets of a $\mathrm{R}\mathrm{C}\mathrm{D}(0,N)$ space with $N\in (1,\mathrm{\infty })$. We also obtain an almost rigidity result of the Talenti-type comparison theorem, whose proof relies on a compactness on varying spaces.