EXISTENCE AND NONEXISTENCE OF SOLUTIONS FOR A HARMONIC EQUATION WITH CRITICAL NONLINEARITY

This paper is concerned with the harmonic equation(P■ε) : ?u = 0, u > 0 in B~n and ?u/?ν+(（n-2）/2)u =(（n-2）/2) Kun/（n-2）■ε on Sn-1where B~n is the unit ball in R~n, n ≥ 4 with Euclidean metric g0, ?B~n= Sn-1is its boundary, K is a function on Sn-1and ε is a small positive parameter. We construct solutions of the subcritical equation(P-ε) which blow up at one critical point of K. We give also a sufficient condition on the function K to ensure the nonexistence of solutions for(P-ε) which blow up at one point. Finally, we prove a nonexistence result of single peaked solutions for the supercritical equation(P+ε).