  We investigate the next Trudinger-Moser critical equations, \[ \begin{cases}
-\Delta u=\lambda ue^{u^2+\alpha|u|^\beta}&\text{ in }B,\\ u=0&\text{ on
}\partial B, \end{cases} \] where $\alpha>0$,
$(\lambda,\beta)\in(0,\infty)\times(0,2)$ and $B\subset \mathbb{R}^2$ is the
unit ball centered at the origin. We classify the asymptotic behavior of energy
bounded sequences of radial solutions. Via the blow--up analysis and a scaling
technique, we deduce the limit profile, energy, and several asymptotic formulas
of concentrating solutions together with precise information of the weak limit.
In particular, we obtain a new necessary condition on the amplitude of the weak
limit at the concentration point. This gives a proof of the conjecture by
Grossi-Mancini-Naimen-Pistoia in 2020 in the radial case. Moreover, in the case
of $\beta\le1$, we show that any sequence carries at most one bubble. This
allows a new proof of the nonexistence of low energy nodal radial solutions for
$(\lambda,\beta)$ in a suitable range. Lastly, we discuss several counterparts
of our classification result. Especially, we prove the existence of a sequence
of solutions which carries multiple bubbles and weakly converges to a
sign-changing solution.
