Critical points of embeddings of H01,n into Orlicz spaces

For a domain Ω ⊂ n embeddings u → exp(α(|u|/u1, n)n/n − 1) of H01,n(Ω) into Orlicz spaces are considered. At the critical exponent α = αn a loss of compactness reminiscent of the Yamabe problem is encountered; however by a result of Carlesson and Chang, if Ω is a ball the best constant for the above embedding is attained. In dimension n = 2 we identify the limiting problem responsible for the lack of compactness at the critical exponent α2 = 4π in the radially symmetric case and establish the existence of extremal functions also for nonsymmetric domains Ω. Moreover, we establish the existence of two branches of critical points of this embedding beyond the critical exponent α2 = 4π. © 2016 L'Association Publications de l'Institut Henri Poincaré