AbstractSharp existence and nonexistence results for positive solutions of quasilinear elliptic equations with critical growth in geodesic balls on spheres are established. The arguments are based on Pohozaev type identities and asymptotic estimates for Emden–Fowler type equations. By means of spherical symmetrization and the concentration‐compactness principle existence and nonexistence results for general domains on spheres are obtained. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)