  For an integer $n \ge 3$ and any positive number $\epsilon$ we establish the
existence of smooth functions K on $R^n \setminus \{0 \}$ with $|K - 1| \le
\epsilon$, such that the equation $\Delta u + n (n - 2) K u^{{n + 2}\over {n -
2}} = 0$ in $R^n \setminus \{0 \}$ has a smooth positive solution which blows
up at the origin (i.e., u does not have slow decay near the origin).
Furthermore, we show that in some cases K can be extended as a Lipschitz
function on ${\R}^n.$ These provide counter-examples to a conjecture of C.-S.
Lin when n > 4, and Taliaferro's conjecture.
