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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Large solutions for Yamabe and similar problems on domains in Riemannian manifolds


Author: Martin Dindoš
Journal: Trans. Amer. Math. Soc. 363 (2011), 5131-5178
MSC (2000): Primary 35J25, 35J60; Secondary 53C21
Published electronically: May 4, 2011
MathSciNet review: 2813411
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Abstract: We present a unified approach to study large positive solutions (i.e., $ u(x)\to\infty$ as $ x\to\partial\Omega$) of the equation $ \Delta u+hu-k\psi(u)=-f$ in an arbitrary domain $ \Omega$. We assume $ \psi(u)$ is convex and grows sufficiently fast as $ u\to\infty$. Equations of this type arise in geometry (Yamabe problem, two dimensional curvature equation) and probability (superdiffusion). We prove that both existence and uniqueness are local properties of points of the boundary $ \partial\Omega$; i.e., they depend only on properties of $ \Omega$ in arbitrarily small neighborhoods of each boundary point. We also find several new necessary and sufficient conditions for existence and uniqueness of large solutions including an existence theorem on domains with fractal boundaries.


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Additional Information

Martin Dindoš
Affiliation: School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Mayfield Road, JCMB KB, Edinburgh EH9 3JZ, United Kingdom

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05282-7
PII: S 0002-9947(2011)05282-7
Keywords: Elliptic equations, large solutions
Received by editor(s): August 1, 2008
Received by editor(s) in revised form: May 30, 2009, and May 31, 2009
Published electronically: May 4, 2011
Additional Notes: The author was supported in part by EPSRC grant EP/F014589/1-253000 RA0347.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.