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Group invariance and Pohozaev identity in Moser-type inequalities. (English) Zbl 1271.46030

Summary: We study the so-called limiting Sobolev cases for embeddings of the spaces \(W_0^{1,n}(\Omega )\), where \(\Omega \subset \mathbb R^n\) 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.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35B65 Smoothness and regularity of solutions to PDEs
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