We study the Dirichlet problem in the unit ball $B_1$ of $\mathbb R^2$ for the Hénon-type equation of the form \begin{equation*} \begin{cases} -\Delta u =\lambda u+|x|^{\alpha}f(u) & \mbox{in } B_1, \\ \quad \ \ u = 0 & \mbox{on } \partial B_1, \end{cases} \end{equation*} where $f(t)$ is a $C^1$-function in the critical growth range motivated by the celebrated Trudinger-Moser inequality. Under suitable hypotheses on constant $\lambda$ and $f(t)$, by variational methods, we study the solvability of this problem in appropriate Sobolev s paces.

中文翻译：

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亚临界和临界指数增长的$ \ mathbb R ^ 2 $中的Hénon椭圆方程

我们针对\\ begin {equation *} \ begin {cases}-\ Delta u = \ lambda u + | x形式的Hénon型方程，研究$ \ mathbb R ^ 2 $的单位球$ B_1 $中的Dirichlet问题| ^ {\ alpha} f（u）＆\ mbox {in} B_1，\\ \ quad \ \ u = 0＆\ mbox {on} \ partial B_1，\ end {cases} \ end {equation *}其中$ f（t）$是著名的Trudinger-Moser不平等所激发的临界增长范围内的$ C ^ 1 $函数。在关于常数$ \ lambda $和$ f（t）$的适当假设下，通过变分方法，我们研究了在适当的Sobolev步伐下该问题的可解性。