NDLI: Beyond the Trudinger-Moser supremum

Content Provider	Springer Nature Link
Author	Pi, Manuel Musso, Monica Ruf, Bernhard
Copyright Year	2011
Abstract	Let Ω be a bounded, smooth domain in $${\mathbb{R}^2}$$ . We consider the functional $$I(u) = \int_\Omega e^{u^2}\,dx$$ in the supercritical Trudinger-Moser regime, i.e. for $${\int_\Omega \|\nabla u\|^2dx > 4\pi}$$ . More precisely, we are looking for critical points of I(u) in the class of functions $${u \in H_0^1 (\Omega )}$$ such that $${\int_\Omega \|\nabla u\|^2 \, dx = 4\, \pi \, k\, (1+\alpha)}$$ , for small α > 0. In particular, we prove the existence of 1-peak critical points of I(u) with $${\int_\Omega \|\nabla u\|^2dx = 4\pi(1 + \alpha)}$$ for any bounded domain Ω, 2-peak critical points with $${\int_\Omega \|\nabla u\|^2dx = 8\pi(1 + \alpha)}$$ for non-simply connected domains Ω, and k-peak critical points with $${\int_\Omega \|\nabla u\|^2 dx = 4k \pi(1 + \alpha)}$$ if Ω is an annulus.
Ending Page	576
Page Count	34
Starting Page	543
File Format	PDF
ISSN	09442669
e-ISSN	14320835
Journal	Calculus of Variations and Partial Differential Equations
Issue Number	3-4
Volume Number	44
Language	English
Publisher	Springer-Verlag
Publisher Date	2011-08-25
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems Systems Theory, Control Calculus of Variations and Optimal Control; Optimization Theoretical, Mathematical and Computational Physics Analysis Second-order elliptic equations
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis

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1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
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4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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