We obtain the local uniqueness of positive solution for equation with inverse square terms in a smooth bounded domain. This is done by analyzing the Green function and the blow up profile of positive solution of some critical exponent problems with Hardy potential in a smooth bounded domain. We also determine the critical constant of the blow up behavior. This extends the result of Grossi  and Cerqueti  and Cao-Peng  to singular operators.
"Uniqueness of positive solutions with critical exponent and inverse square potential." Adv. Differential Equations 19 (9/10) 879 - 910, September/October 2014.