  We are concerned with the Lane-Emden problem \begin{equation*} \begin{cases}
-\Delta u=u^{p} &{\text{in}~\Omega},\\[0.5mm]
  u>0 &{\text{in}~\Omega},\\[0.5mm] u=0 &{\text{on}~\partial \Omega},
\end{cases} \end{equation*} where $\Omega\subset \mathbb R^2$ is a smooth
bounded domain and $p>1$ is sufficiently large. Improving some known asymptotic
estimates on the solutions, we prove the non-degeneracy and local uniqueness of
the multi-spikes positive solutions for general domains. Our methods mainly use
ODE's theory, various local Pohozaev identities, blow-up analysis and the
properties of Green's function.
