Abstract
The aim of this paper is to study symmetry and monotonicity of positive solutions for the following overdetermined problem
where
, p>0, A>0,
,
is a bounded domain. We first prove that
on
if and only if Ω is a ball. Next we consider the partially overdetermined problem. If Γ is a proper open set of
and u = C in
, we show that under some assumptions on the geometry of Γ, Ω is a ball. Furthermore, we derive that all positive solutions of above equations are radially symmetric and monotone increasing with respect to the radius by using the method of moving planes.
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Acknowledgements
The authors thank Prof. Wenxiong Chen for valuable discussions and encouragement on this manuscript. This work was completed when the first author was visiting the Department of Mathematical Science, Yeshiva University. The first author appreciates the hospitality of Prof. Wenxiong Chen.
Disclosure statement
No potential conflict of interest was reported by the authors.