We establish the uniqueness and
the blow-up rate of the large positive solution of the singular
boundary problem $-\Delta_{p} u=\lambda u^{p-1}-b(x) u^q$ in
$B_R(x_0)$, $u|_{\partial B_R(x_0)}=+\infty$, where $B_R(x_0)$ is
a ball domain of radius $R$ centered at $x_0\in\mathbb{R}^N$,
$N\geq3$, $\lambda>0$ and the potential function $b$ is a positive
radially symmetric function. Our result extends the previous work
by Ouyang and Xie from the case $p=2$ to the case $p>2$ and we
prove that any large solution $u$ must satisfy
$$\lim_{d(x)\rightarrow 0}\frac{u(x)}{K(b^{*}(\|x-x_{0}\|))^{-\beta}}=1,$$
where $d(x)= {\rm dist}(x,
\partial B_{R}(x_{0}))$,
$K$ is a constant defined by
$$K:=\left[(p-1)[(\beta +1)C_{0}-1]\beta^{p-1}(C_{0}b_{0})^{(p-2)/2}\right]^{\frac{1}{q-p+1}},$$
with $$\beta:=\frac{p}{2(q-p+1)},\;q>p-1>1,\; b_{0}:=b(R)>0,\;
C_{0}:=\lim_{r\rightarrow R}\frac{(B(r))^{2}}{b^{*}(r)b(r)}\geq
1$$ and
$$B(r):=\int_{r}^{R}
b(s)ds,\; b^{*}(r)=\int_{r}^{R}\int_{s}^{R} b(t)dt ds.$$