  In this paper, we prove that there exists at most one positive radial weak
solution to the following quasilinear elliptic equation with singular critical
growth \[ \begin{cases} -\Delta_{p}u-{\displaystyle
\frac{\mu}{|x|^{p}}|u|^{p-2}u}{\displaystyle
=\frac{|u|^{\frac{(N-s)p}{N-p}-2}u}{|x|^{s}}}+\lambda|u|^{p-2}u & \text{in
}B,\\ u=0 & \text{on }\partial B, \end{cases} \] where $B$ is an open finite
ball in $\mathbb{R}^{N}$ centered at the origin, $1<p<N$,
$-\infty<\mu<((N-p)/p)^{p}$, $0\le s<p$ and $\lambda\in\mathbb{R}$. A related
limiting problem is also considered.
