  We consider positive solution to the weighted elliptic problem
\begin{equation*} \left \{ \begin{array}{ll} -{\rm div} (|x|^\theta \nabla
u)=|x|^\ell u^p \;\;\; \mbox{in $\mathbb{R}^N \backslash {\overline B}$},\\ u=0
\;\;\; \mbox{on $\partial B$}, \end{array} \right. \end{equation*} where $B$ is
the standard unit ball of $\mathbb{R}^N$. We give a complete answer for the
existence question when $N':=N+\theta>2$. In particular, for $N' > 2$ and
$\tau:=\ell-\theta >-2$, it is shown that the problem admits a unique positive
radial solution for $p>p_s:=\frac{N'+2+2\tau}{N'-2}$, while for any $ 0<p \leq
p_s$, the only nonnegative solution is $u \equiv 0$.
