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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

##### MSC:
 35J60 Nonlinear elliptic equations 58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals 35J20 Variational methods for second-order elliptic equations
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##### References:
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