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Large blow-up sets for the prescribed Q-curvature equation in the Euclidean space

Hyder, Ali and Iula, Stefano and Martinazzi, Luca. (2016) Large blow-up sets for the prescribed Q-curvature equation in the Euclidean space. Preprints Fachbereich Mathematik, 2016 (21).

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Abstract

Let $m\ge 2$ be an integer. For any open domain $\Omega\subset\mathbb{R}^{2m}$, non-positive function $\varphi\in C^{\infty}(\Omega)$ such that $\Delta^m \varphi \equiv 0$ and bounded sequence $(V_k) \subset L^\infty (\Omega)$ we prove the existence of a sequence of functions $(u_k) \subset C^{2m-1}(\Omega)$ solving the Liouville equation of order 2m
\[
(-\Delta)^m u_k = V_k e^{2m u_k} in \Omega, \limsup_{k\to\infty} \int_{\Omega} e^{2m u_k} dx < \infty,
\]
and blowing up exactly on the set $S_{\varphi} := \{x\in\Omega : \varphi(x) = 0\}$, i.e.
\[
\lim_{k\to\infty} u_k(x) = +\infty for x\in S_{\varphi} and \lim_{k\to\infty} u_k(x) = -\infty for x\in\Omega\setminus S_{\varphi},
\]
thus showing that a result of Adimurthi, Robert and Struwe is sharp. We extend this result to the boundary of $\Omega$ and to the case $\Omega = \mathbb{R}^{2m}$. Several related problems remain open.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Ehemalige Einheiten Mathematik & Informatik > Analysis (Martinazzi)
12 Special Collections > Preprints Fachbereich Mathematik
UniBasel Contributors:Hyder, Ali and Iula, Stefano and Martinazzi, Luca
Item Type:Preprint
Publisher:Universität Basel
Language:English
Last Modified:06 May 2019 17:59
Deposited On:28 Mar 2019 09:51

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