  We study the existence and nonexistence of positive (super) solutions to the
nonlinear $p$-Laplace equation $$-\Delta_p
u-\frac{\mu}{|x|^p}u^{p-1}=\frac{C}{|x|^{\sigma}}u^q$$ in exterior domains of
${\R}^N$ ($N\ge 2$). Here $p\in(1,+\infty)$ and $\mu\le C_H$, where $C_H$ is
the critical Hardy constant. We provide a sharp characterization of the set of
$(q,\sigma)\in\R^2$ such that the equation has no positive (super) solutions.
  The proofs are based on the explicit construction of appropriate barriers and
involve the analysis of asymptotic behavior of super-harmonic functions
associated to the $p$-Laplace operator with Hardy-type potentials, comparison
principles and an improved version of Hardy's inequality in exterior domains.
In the context of the $p$-Laplacian we establish the existence and asymptotic
behavior of the harmonic functions by means of the generalized
Pr\"ufer-Transformation.
