On the Extremal Functions of Sobolev-poincaré Inequality

where ua = 1 vol(Mn) ∫ Mn u, p∗ = np/(n − p) is the Sobolev conjugate of p. This inequality can be proved by combining Sobolev inequality with Poincaré inequality, see, for example, Hebey’s book [8]. In this paper we are interested in the estimates of the best constant and the existence of extremal functions to the above inequality. Analytically these are naturally motivated questions. On the other hand they may have interesting geometric implications, in particular, in the study of Poincaré’s isoperimetric inequality. We shall discuss this geometric issue in another paper [9]. Generally speaking, the existence of extremal functions is not a trivial issue, given the fact that p∗ is the critical Sobolev exponent for the Sobolev embedding theorem. In this paper, for a model manifold -the unit sphere Sn with the standard metric g0 = ∑n+1 i=1 dx 2 i , we obtain a fairly satisfying result. Define the Sobolev-Poincaré quotient by