Abstract: Let $\Omega$ denote a smooth, bounded domain in $\mathbb{R}^N (N\geq 2)$. Suppose that $g$ is a nondecreasing $C^1$ positive function and assume that $b(x)$ is continuous and nonnegative in $\Omega$, and that it may be singular on $\partial\Omega$. In this paper, we provide sufficient and necessary conditions on the existence of boundary blow-up solutions to the $p$-Laplacian problem
$\Delta_p u=b(x)g(u) \mbox{ for } x \in \Omega,\; u(x)\rightarrow +\infty \mbox{ as } { dist}(x,\partial \Omega)\rightarrow 0$.
The estimates of such solutions are also investigated. Moreover, when $b$ has strong singularity, the nonexistence of boundary blow-up (radial) solutions and infinitely many radial solutions are also considered.

Key words: singular $p$-Laplacian equation, boundary blow-up, sub-supersolution method, existence, nonexistence and estimates, sufficient and necessary conditions