Radial symmetry for p-harmonic functions in exterior and punctured domains

We prove symmetry for the p-capacitary potential satisfying under Serrin's overdetermined condition Here is star-shaped. Our proof uses the maximum principle for an appropriate P-function, some integral identities, the isoperimetric inequality, and a Soap Bubble-type Theorem. We then treat the case , improving previous results present in the literature. Finally, with analogous tools, we give a new proof of symmetry for the interior overdetermined problem <g<g in a bounded star-shaped domain .