We investigate the next Trudinger–Moser critical equations,

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 et al. (Math Ann, to appear) in 2020 in the radial case. Moreover, in the case of \(\beta \le 1\), 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.

我们研究下一个Trudinger-Moser临界方程，

其中\（\ alpha> 0 \），\（（\ lambda，\ beta）\ in（0，\ infty）\ times（0,2）\）和\（B \ subset {\ mathbb {R}} ^ 2 \）是以原点为中心的单位球。我们将径向解的能量有界序列的渐近行为分类。通过爆炸分析和定标技术，我们推导了极限曲线，能量和集中解的几个渐近公式以及弱极限的精确信息。特别是，我们在集中点的弱极限幅度上获得了一个新的必要条件。这证明了Grossi等人的猜想。（Math Ann，出现）在2020年出现在放射状情况下。此外，在\（\ beta \ le 1 \）的情况下，我们证明任何序列最多携带一个气泡。这为在适当范围内\（（\ lambda，\ beta）\）的低能节点径向解不存在提供了新的证明。最后，我们讨论分类结果的几个对应项。特别是，我们证明了一系列解的存在，这些解带有多个气泡并且弱收敛到一个符号转换解。