We consider the Hénon equation(P_α)$-\mathrm{\Delta }u=|x{|}^{\alpha }|u{|}^{p-1}u\mathrm{\text{in}}{B}^N,u=0\mathrm{\text{on}}\partial B^N,$ where $B^N\subset {\mathbb{R}}^N$ is the open unit ball centered at the origin, $N\ge 3$, $p>1$ and $\alpha >0$ is a parameter. We show that, after a suitable rescaling, the two-dimensional Lane-Emden equation$-\mathrm{\Delta }w=|w{|}^{p-1}w\text{in}{B}^2,w=0\text{on}\partial B^2,$ where $B^2\subset {\mathbb{R}}^2$ is the open unit ball, is the limit problem of ($P_{\alpha }$), as $\alpha \to \infty$, in the framework of radial solutions. We exploit this fact to prove several qualitative results on the radial solutions of ($P_{\alpha }$) with any fixed number of nodal sets: asymptotic estimates on the Morse indices along with their monotonicity with respect to α; asymptotic convergence of their zeros; blow up of the local extrema and on compact sets of $B^N$. All these results are proved for both positive and nodal solutions.

我们认为期Hénon方程（P _α）$-\mathrm{\Delta }ü=|X{|}^{\alpha }|ü{|}^{p-\mathrm{1个}}ü\mathrm{\text{在}}{乙}^{ñ}，ü=0\mathrm{\text{上}}\partial {乙}^{ñ}，$ 在哪里 ${乙}^{ñ}\subset {\mathbb{[R}}^{ñ}$ 是以原点为中心的开放式单位球， $ñ\ge 3$， $p>\mathrm{1个}$ 和 $\alpha >0$是一个参数。我们表明，经过适当的缩放后，二维Lane-Emden方程$-\mathrm{\Delta }w=|w{|}^{p-\mathrm{1个}}w\text{在}{乙}^{\mathrm{2个}}，w=0\text{上}\partial {乙}^{\mathrm{2个}}，$ 在哪里 ${乙}^{\mathrm{2个}}\subset {\mathbb{[R}}^{\mathrm{2个}}$ 是开式单位球，是极限问题$P_{\alpha }$）， 作为 $\alpha \to \infty$，在径向解的框架中。我们利用这一事实证明了（）的径向解的几个定性结果$P_{\alpha }$）具有任意数量的节点集：摩尔斯指数的渐近估计以及它们相对于α的单调性；零点的渐近收敛；炸毁局部极值并紧紧套上${乙}^{ñ}$。所有这些结果在正解和节点解中都得到了证明。