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The aim of this paper is to study symmetry and monotonicity of positive solutions for the following overdetermined problemwhere,  p &gt;0,  A &gt;0,,is a bounded domain. We first prove thatonif and only if Ω is a ball. Next we consider the partially overdetermined problem. If Γ is a proper open set ofand  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.

Overdetermined problems for negative power integral equations on bounded domain