Singular elliptic equation involving the GJMS operator on the standard unit sphere

Abstract

Given a Riemannian compact manifold (M,g) of dimension n>4, we have proven in [1] under some conditions that the equation : Pg(u) = Bu +Au2+Cu (1) where Pg is the GJMS-operator, n = dim(M) > 2k, A, B and C are smooth positive functions on M, p > 1 and 2] denotes the critical Sobolev admits twodistinct positive solutions. The proof of this result is essentially based on the given smooth function ' > 0 with norm k'kPg = 1 fulfilling some conditions ( see Theorem 3 in [1]). In this note we construct an example of such function on the unit standard sphere (Sn; h). Con- sequently the conditions of the Theorem are improved in the case of (Sn; h)