Group invariance and Pohozaev identity in Moser type inequalities

We study the so-called limiting Sobolev cases for embeddings of the spaces $W^{1,n}_0(\Omega)$, where $\Omega\subset\R^n$ is a bounded domain. Differently from J. Moser, we consider optimal embeddings into Zygmund spaces: we derive related Euler-Lagrange equations, and show that Moser's concentrating sequences are the solutions of these equations and thus realize the best constants of the corresponding embedding inequalities. Furthermore, we exhibit a group invariance, and show that Moser's sequence is generated by this group invariance and that the solutions of the limiting equation are unique up to this invariance. Finally, we derive a Pohozaev-type identity, and use it to prove that equations related to perturbed optimal embeddings do not have solutions.