Planar Hénon-type equation with Trudinger-Moser critical growth

This paper focuses on the study of radially symmetric solutions for the planar Hénon-type equation with Trudinger-Moser critical growth$ \begin{equation*} \begin{cases} -\Delta u=\lambda u+|x|^{\alpha}f(u), \;\;& \mbox {in} \ \ B_1,\ u= 0, \;\;& \mbox {on} \ \ \partial B_1, \end{cases} \end{equation*} $in the cases $ 0\le \lambda<\lambda_1 $ and $ \lambda_k<\lambda<\lambda_{k+1} $, respectively, where $ \alpha>-2 $, $ B_1:=\{x\in \mathbb{R}^2 : |x|<1\} $, $ f(t) $ has critical exponential growth at infinity behaviors as $ e^{\beta_0t^2} $ and $ 0<\lambda_1<\lambda_2\le \lambda_3\ldots $ are defined by the eigenvalues of $ (-\Delta, H_{0,\mathrm{rad}}^1(B_1)) $. Especially, we develop a straightforward method with some delicate estimates to determine a fine upper bound for the minimax level (Mountain Pass minimax level if $ 0\le \lambda<\lambda_1 $ and Link minimax level if $ \lambda_k<\lambda<\lambda_{k+1} $), obviously weakening the previous hypothesis on $ \liminf_{t\to\infty}\frac{tf(t)}{e^{\beta_0 t^2}} $. We emphasize that this method is more convenient and applicable for other elliptic problems with critical exponential growth. The results we established extend and improve theones of de Figueiredo-Miyagaki-Ruf [Calc. Var. Partial Differ. Equ. 3: 139-153, 1995] and of de Figueiredo-do Ó-Ruf [Comm. Pure Appl. Math. 55: 135-152, 2002], and are new even for the case that $ \alpha=0 $.