本文研究形如 $$\mathrm{△}(({\mathrm{△}}^{n}u{)}^{(p-1{)}^{\ast }})=f(|x|,u,|\mathrm{\nabla }u|)u^{-\beta },\text{q}x\in {\text{boldmath }R}^N$$ 的奇异非线性多调和方程在 ${\text{boldmath }R}^N$ 上的正整体解, 此处 $p>1$, $\beta \ge 0$ 是常数, $n$ 是自然数, $f:{\overline{R}}_{+}\times R_{+}\times {\overline{R}}_{+}\to {\overline{R}}_{+}$ 是一个连续函数, ${\xi }^{{\delta }^{\ast }}:=\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\xi )\cdot |\xi {|}^{\delta }$, $\xi \in \text{boldmath }R$, $\delta >0$, 给出了该类方程具有无穷多个 其渐进阶刚好为 $|x{|}^{2n}$ 的正整体解的充分条件与必要条件. 这些结论可以推广到更一般的方程.

In this paper N-dimensional singular, nonlinear poly-harmonic equations of the following form $$\mathrm{△}(({\mathrm{△}}^{n}u{)}^{(p-1{)}^{\ast }})=f(|x|,u,|\mathrm{△}u|)u^{-\beta },\text{q}x\in {\text{boldmath }R}^N$$ are considered, where $p>1$, $\beta \ge 0$, $n$ is an integer ($n\ge 1$), ${\xi }^{{\delta }^{\ast }}:=\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\xi )\cdot |\xi {|}^{\delta }$, $\xi \in \text{boldmath }R$, $\delta >0$. and $f:{\overline{R}}_{+}\times R_{+}\times {\overline{R}}_{+}\to R_{+}$ is a continuous function. Some sufficient conditions and necessary conditions are obtained for the existence of infinitely many positive symmetric entire solutions which are asymptotic to positive constant multiples of $|x{|}^{2n}$ as $|x|\to \mathrm{\infty }$. The results can be extended to certain equations of more general form.