Uniqueness of extremals for some sharp Poincaré-Sobolev constants

We study the sharp constant for the embedding of W 1,p 0 (omega) into Lq(omega), in the case 2 < p < q. We prove that for smooth connected sets, when q > p and q is sufficiently close to p, extremal functions attaining the sharp constant are unique, up to a multiplicative constant. This in turn gives the uniqueness of solutions with minimal energy to the Lane-Emden equation, with super-homogeneous right-hand side. The result is achieved by suitably adapting a linearization argument due to C.-S. Lin. We rely on some fine estimates for solutions of p-Laplace-type equations by L. Damascelli and B. Sciunzi.