Blow-up solutions of nonlinear elliptic equations in Rn with critical exponent

For an integer ngreater than or equal to3 and any positive number epsilon, we establish the existence of smooth functions K on R-n\{0} with |K-1|less than or equal toepsilon, such that the equation Deltau + n (n - 2) Ku n + 2/n - 2 = 0 in R-n\{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 situations 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 a question of Taliaferro.