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

Martinazzi, Luca

In: Calculus of Variations and Partial Differential Equations, 2009, vol. 36, no. 4, p. 493-506

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    Title
    • A threshold phenomenon for embeddings of $${H^m_0}$$ into Orlicz spaces
    Author
    • Martinazzi, Luca. ETH Zurich, Rämistrasse 101, 8092, Zurich, Switzerland
    Document Type
    • Postprint
    Language
    • English
    Published in
    • Calculus of Variations and Partial Differential Equations, 2009, vol. 36, no. 4, p. 493-506. Springer-Verlag
    Other Version
    Classification
    • Mathematics
    OAI-PMH Identifier
    • oai:doc.rero.ch:321546
    Summary
    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 λ k → 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(S 2m ) is the total Q-curvature of S 2m