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.
Categories: math.AP math.DG