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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 theunit ball centered at the origin. We classify the asymptotic behavior of energybounded sequences of radial solutions. Via the blow--up analysis and a scalingtechnique, we deduce the limit profile, energy, and several asymptotic formulasof concentrating solutions together with precise information of the weak limit.In particular, we obtain a new necessary condition on the amplitude of the weaklimit at the concentration point. This gives a proof of the conjecture byGrossi-Mancini-Naimen-Pistoia in 2020 in the radial case. Moreover, in the caseof $\beta\le1$, we show that any sequence carries at most one bubble. Thisallows a new proof of the nonexistence of low energy nodal radial solutions for$(\lambda,\beta)$ in a suitable range. Lastly, we discuss several counterpartsof our classification result. Especially, we prove the existence of a sequenceof solutions which carries multiple bubbles and weakly converges to asign-changing solution.

Concentration profile, energy, and weak limits of radial solutions to semilinear elliptic equations with Trudinger-Moser critical nonlinearities