We consider the uniqueness of the following positive solutions of anisotropic elliptic equation: \begin{equation}\nonumber \left\{ \begin{aligned} -Δ^F _p u&=u^q \quad \text{in} \quad Ω, u&=0 \quad \text{on} \quad \partial Ω, \end{aligned} \right. \end{equation} where $p>\frac{3}{2}$ is a constant. We utilize the linearized method to derive the uniqueness results, which extends the conclusion obtained by L. Brasco and E. Lindgren.