We are interested in singular positive solutions of a quasilinear elliptic equation with a singular coefficient $r^{-(\gamma -1)}{\left(r^{\alpha }|u^{\prime }{|}^{\beta -1}{u}^{\prime }\right)}^{\prime }+k{r}^{\alpha -\beta -\gamma }{u}^{\beta }+u^p=0,0<r<\mathrm{\infty },$ where $1\le \beta <\alpha <\gamma$, $p>\beta$, $0<k<(\alpha -\beta -\beta q)q^{\beta }$ and $q:=(-\alpha +\beta +\gamma )/(p-\beta )$. The differential operator includes the standard Laplace, m-Laplace and ℓ-Hessian operators. In the case $(\alpha ,\beta ,\gamma )=(N-1,1,N+\delta )$, a solution of the problem gives a radial solution of $\mathrm{\Delta }u+ku/|x{|}^2+|x{|}^{\delta }{u}^p=0$. This problem has a one-parameter family of singular positive solutions and one exact singular solution. We obtain an intersection number of two singular solutions in the critical and supercritical cases. The main technical tool is a phase plane analysis.

我们对具有奇异系数的拟线性椭圆方程的奇异正解感兴趣$r^{-(\gamma -1)}{\left(r^{\alpha }|{你}^{\text{'}}{|}^{\beta -1}{你}^{\text{'}}\right)}^{\text{'}}+ķr^{\alpha -\beta -\gamma }{你}^{\beta }+{你}^p=0,0<r<\mathrm{\infty },$在哪里$1\le \beta <\alpha <\gamma$,$p>\beta$,$0<ķ<(\alpha -\beta -\beta q)q^{\beta }$和$q:=(-\alpha +\beta +\gamma )/(p-\beta )$. 微分算子包括标准的拉普拉斯算子、m -Laplace 算子和 ℓ-Hessian 算子。在这种情况下$(\alpha ,\beta ,\gamma )=(ñ-1,1,ñ+\delta )$, 问题的一个解给出一个径向解$\mathrm{\Delta }你+ķ你/|X{|}^2+|X{|}^{\delta }{你}^p=0$. 这个问题有一个单参数的奇异正解和一个精确奇异解。我们在临界和超临界情况下获得了两个奇异解的交集数。主要的技术工具是相平面分析。