  We are concerned with the Moser-Trudinger problem \begin{equation*}
\begin{cases} -\Delta u=\lambda ue^{u^2}~~&\mbox{in}~\Omega,\\[0.5mm] u>0 ~~
&{\text{in}~\Omega},\\[0.5mm] u=0~~&\mbox{on}~\partial \Omega, \end{cases}
\end{equation*} where $\Omega \subset \mathbb{R}^2$ is a smooth bounded domain
and $\lambda>0$ is sufficiently small. Qualitative analysis for Moser-Trudinger
nonlinearities has been studied in recent decades, however there is still a lot
of clarity about this issue, even for a low energy. The reason is that this
problem is a critical exponent for dimension two and will lose compactness.
Here by using a variety of local Pohozaev identities, we qualitatively analyze
the positive solutions of Moser-Trudinger problem with a low energy, which
contains the Morse index, non-degeneracy, asymptotic behavior, uniqueness and
symmetry of solutions. Since the fundamental solution of $-\Delta$ in $\Omega
\subset \mathbb{R}^2$ is in logarithmic form and the corresponding bubble is
exponential growth, more precise asymptotic behavior of the solutions is
needed, which is of independent interest. Moreover, to obtain our results, some
ODE's theory will be used to a prior estimate of the solutions and some
elliptic theory in dimension two will play a crucial role.
