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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