Existence and nonexistence of solutions for a harmonic equation with critical nonlinearity

This paper is concerned with the harmonic equation ( P∓ɛP∓ɛ): ‡u = 0, u   < 0 in BnBn and ∂u∂v+n-22u=n-22Kunn-2∓ɛ on Sn-1Sn-1 where BnBn is the unit ball in ℝnℝn, n ≥ 4 with Euclidean metric g0, ∂Bn=Sn-1∂Bn=Sn-1 is its boundary, K   is a function on Sn-1Sn-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)