Symmetry in exterior boundary value problems for quasilinear elliptic equations via blow-up and a priori estimates. (English) Zbl 0951.35045

The paper is concerned with the \(p\)-capacitary problem: \[ \text{div}(|Du|^{p-2}Du)=0\text{ in }\Omega =\mathbb{R}^{n}\backslash \overline{\Omega }_{1},\quad u=1\text{ on }\partial \Omega _{1},\;u\rightarrow 0\text{ as } |x|\rightarrow \infty ,\tag{1} \] where \(1<p<n\) and \(\Omega _{1}\) is a connected, bounded open set, starlike with respect to the origin which is assumed to belong to \(\Omega _{1}\). For the weak solutions of (1) three main properties of the function \[ P(u,x)=\frac{|Du(x)|^{p}}{u(x)^{\frac{p(n-1)}{n-p}}} \] are studied: \(i)\) a strong maximum principle; \(ii)\) the maximum value of \(P\) in the spherically symmetric configuration of \((1)\), and \(iii)\) a rescaling property in the sense that \[ P(r^{\frac{n-p}{p-1}}u(rx),x)=P(u,rx). \] As a consequence, it is given a new proof of a recent result of W. Reichel [Z. Anal. Anwend. 15, 619-635 (1996; Zbl 0857.35010)], concerning the spherical symmetry in (1) with the overdetermined boundary condition \(|Du|=c>0\) on \(\partial \Omega _{1}\). In order to prove some of the main results the authors exploited besides two Serrin’s results: J. Serrin [Acta Math. 111, 247-302 (1964; Zbl 0128.09101)], and J. Serrin [Acta Math. 113, 219-240 (1965; Zbl 0173.39202)], the local regularity theory for uniformly elliptic operators used from the book of O. Ladyzhenskaya and N. Ural’tseva [Linear and quasilinear elliptic equations (Mathematics in Science and Engineering 46, New York-London: Academic Press. XVIII) (1968; Zbl 0164.13002)], and A. D. Alexandrov’s well known theorem on the characterisation of spheres.


35J65 Nonlinear boundary value problems for linear elliptic equations
31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B50 Maximum principles in context of PDEs