There is a $1$--parameter family of inequalities where the sharp constant and extremal functions are known. I believe this was first established by Del Pino and Dolbeault. Cordero, Nazaret, and Villani gave a beautiful optimal transportation proof. See their paper for the relevant references. The family includes the sharp Sobolev and sharp log-Sobolev inequalities.

There is a $1$--parameter family of inequalities where the sharp constants and corresponding extremal functions are known. I believe this was first established by Del Pino and Dolbeault. Cordero, Nazaret, and Villani gave a beautiful optimal transportation proof. See their paper for the relevant references. The family includes the sharp Sobolev and sharp log-Sobolev inequalities.

ADDED: Here are the exactly references:

Del Pino, Manuel(RCH-UCSP-EM); Dolbeault, Jean(F-PARIS9-A) Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions. J. Math. Pures Appl. (9) 81 (2002), no. 9, 847–875.

Cordero-Erausquin, D.(F-MARN-AMA); Nazaret, B.(F-ENSLY-PM); Villani, C.(F-ENSLY-PM) A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. Math. 182 (2004), no. 2, 307–332.