  We consider the uniqueness of the following positive solutions of
$m$-Laplacian equation: \begin{equation} \left\{ \begin{aligned} -\Delta _m
u&=\lambda u^{m-1}+u^{p-1} \quad \text{in} \quad \Omega\\ u&=0 \quad \text{on}
\quad \partial \Omega \end{aligned} \right. \qquad(0.1)\end{equation} where
$m>1$ is a constant. When $p\rightarrow m$, the uniqueness of positive
solutions of $(0.1)$ is shown which is based on the essential uniqueness of
first eigenfunction for $m$-Laplacian equation. Futhermore, we also prove the
uniqueness results when $(0.1)$ is a perturbation of Laplacian equation.
