Symmetry in exterior boundary value problems for quasilinear elliptic equations via blow-up and a priori estimates

Abstract

Given a connected, bounded open set $\Omega_1 \subset \mathbb{R}^n$, we use a maximum principle, and compactness arguments to study the properties of the function $P(u,x)$ in (1.5) below associated to a weak solution of the exterior $p$-capacitary problem, \[ \hbox{\rm div\,}(|Du|^{p-2}Du) = 0\ \text{in} \ \Omega=\mathbb{R}^n \setminus \overline{\Omega_1}, \quad \quad 1<p<n, \] $u=1$ on $\partial{\Omega_1}$, $u(x)\to 0$ as $|x|\to \infty$. As a consequence of our results we prove spherical symmetry for the solution $u$ and for the condenser $\Omega_1$ when the overdetermined boundary condition $|Du|=c>0$ on $\partial\Omega_1$ is imposed. This provides a new proof of a recent result of Reichel \cite{31}.