摘要
This paper is concerned with the harmonic equation(P;) : ?u = 0, u > 0 in B;and ?u/?ν+(（n-2）/2)u =(（n-2）/2) Ku;on S;where B;is the unit ball in R;, n ≥ 4 with Euclidean metric g;, ?B;= S;is its boundary, K is a function on S;and ε is a small positive parameter. We construct solutions of the subcritical equation(P;) 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;) which blow up at one point. Finally, we prove a nonexistence result of single peaked solutions for the supercritical equation(P;).
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) Ku^(n/(n-2)■ε) on S^(n-1)where B^n is the unit ball in R^n, n ≥ 4 with Euclidean metric g_0, ?B^n= S^(n-1)is its boundary, K is a function on S^(n-1)and 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)).
基金
the Deanship of Scientific Research at Taibah University on material and moral support in the financing of this research project
