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Critical points of embeddings of \(H_ 0^{1,n}\) into Orlicz spaces. (English) Zbl 0664.35022
From the author’s abstract: For a domain \(\Omega \subset {\mathbb{R}}^ n\) embeddings \(u\to \exp (\alpha (| u| /\| u\|_{1,n})^{n/n-1})\) of \(H_ 0^{1,n}(\Omega)\) into Orlicz spaces are considered. At the critical exponent \(\alpha =\alpha_ n\) a loss of compactness reminiscent of the Yamabe problem is encountered; however, by a result of Carleson and Chang, if \(\Omega\) is a ball the best constant for the above embedding is attained.
In dimension \(n=2\), the author identifies the “limiting problem” responsible for the lack of compactness at the critical exponent \(\alpha_ 2=4\pi\) in the radially symmetric case and establishes the existence of extremal functions also for nonsymmetric domains \(\Omega\). Moreover, he establishes the existence of two “branches” of critical points of this embedding beyond the critical exponent \(\alpha_ 2=4\pi\). References include 16 items.
Reviewer: J.E.Bouillet

35J60 Nonlinear elliptic equations
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
35J20 Variational methods for second-order elliptic equations
Full Text: DOI Numdam EuDML
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