设$p>1$, $\beta \ge 0$是常数,$n$是自然数, $f:{\overline{R}}_{+}\times R_{+}\times {\overline{R}}_{+}\to R$ 是一个连续函数. 本文研究形如 $\mathrm{△}(|{\mathrm{△}}^{n}u{|}^{p-2}({\mathrm{△}}^{n}u))=f(|x|,u,|\mathrm{\nabla }u|)u^{-\beta },x\in R^N(N\ge 3)$ 的奇异非线性$p$-调和方程的正整体解,给出了该类方程具有无穷多个其渐近阶刚好为 $|x{|}^{(2n-2)}$（当$|x|\to \mathrm{\infty }$时）的径向对称的正整体解的若干充分条件.

Suppose $p>1,\beta \ge 0$ are constants, $n$ is a positive integer and $f:{\overline{R}}_{+}\times R_{+}\times {\overline{R}}_{+}\to R$ is a continuous function. This paper investigates $N$-dimensional singular, nonlinear $p$-harmonic equations of the form $$\mathrm{\Delta }(|{\mathrm{\Delta }}^{n}u{|}^{p-2}({\mathrm{\Delta }}^{n}u))=f(|x|,u,|\mathrm{\nabla }u|)u^{-\beta },\text{q}x\in R^N(N\ge 3),$$ and gives some sufficient conditions under which the equations have infinitely many radial positive entire solutions which are asymptotic to positive constant multiples of $|x{|}^{(2n-2)}$ as $|x|\to \mathrm{\infty }$.