Existence and nonexistence results for a weighted elliptic equation in exterior domains

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 ${\mathbb{R}}^N\mathrm{\backslash }\bar{B}$},\ u=0 ;;; \mbox{on $\mathrm{\partial }B$}, \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 $N^{\mathrm{\prime }}:=N+\theta >2$. In particular, for $N^{\mathrm{\prime }}>2$ and $\tau :=\mathrm{\ell }-\theta >-2$, it is shown that the problem admits a unique positive radial solution for $p>p_s:=\frac{N^{\mathrm{\prime }}+2+2\tau }{N^{\mathrm{\prime }}-2}$, while for any $ 0<p \leq p_s$,theonlynonnegativesolutionis$u \equiv 0$.