A threshold phenomenon for embeddings of {H^m

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A threshold phenomenon for embeddings of {H^m_0} into Orlicz spaces

JournalArticle (Originalarbeit in einer wissenschaftlichen Zeitschrift)

ID	2310184
Author(s)	Martinazzi, Luca
Author(s) at UniBasel	Martinazzi, Luca
Year	2009
Title	A threshold phenomenon for embeddings of {H^m_0} into Orlicz spaces
Journal	Calculus of Variations and Partial Differential Equations
Volume	36
Number	4
Pages / Article-Number	493-506
Abstract	Given an open bounded domain {\Omega\subset\mathbb {R}^{2m}} with smooth boundary, we consider a sequence {(u_k)_{k\in\mathbb{N}}} of positive smooth solutions to \left\{\begin{array}{ll} (-\Delta)^m u_k=\lambda_k u_k e^{mu_k^2} \quad\quad\quad\quad\quad {\rm in}\,\Omega\\ u_k=\partial_\nu u_k=\cdots =\partial_\nu^{m-1} u_k=0 \quad {\rm on }\, \partial \Omega, \end{array}\right. where λk → 0+. Assuming that the sequence is bounded in {H^m_0(\Omega)} , we study its blow-up behavior. We show that if the sequence is not precompact, then \liminf_{k\to\infty}\\|u_k\\|^2_{H^m_0}:=\liminf_{k\to\infty}\int\limits_\Omega u_k(-\Delta)^m u_k dx\geq \Lambda_1, where Λ1 = (2m − 1)!vol(S2m) is the total Q-curvature of S2m.
Publisher	Springer
ISSN/ISBN	0944-2669 ; 1432-0835
edoc-URL	http://edoc.unibas.ch/49896/
Full Text on edoc	No
Digital Object Identifier DOI	10.1007/s00526-009-0239-0

17/04/2024

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