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

 

Positive super-solutions to semi-linear second-order non-divergence type elliptic equations in exterior domains


Authors: Vladimir Kondratiev, Vitali Liskevich and Zeev Sobol
Journal: Trans. Amer. Math. Soc. 361 (2009), 697-713
MSC (2000): Primary 35J60, 35B33; Secondary 35B05
Published electronically: September 26, 2008
MathSciNet review: 2452821
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Abstract: We study the problem of the existence and non-existence of positive super-solutions to a semi-linear second-order non-divergence type elliptic equation $ \sum_{i,j=1}^N a_{ij}(x)\frac{\partial ^2 u}{\partial x_i \partial x_j}+u^p=0$, $ -\infty<p<\infty$, with measurable coefficients in exterior domains of $ \mathbb{R}^N$. We prove that in a ``generic'' situation there is one critical value of $ p$ that separates the existence region from non-existence. We reveal the quantity responsible for the qualitative picture and for the numerical value of the critical exponent which becomes available under a mild stabilization condition at infinity.


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Additional Information

Vladimir Kondratiev
Affiliation: Department of Mathematics and Mechanics, Moscow State University, Moscow 119 899, Russia
Email: kondrat@vnmok.math.msu.su

Vitali Liskevich
Affiliation: Department of Mathematics, Swansea University, Swansea SA2 8PP, United Kingdom
Email: V.A.Liskevich@swansea.ac.uk

Zeev Sobol
Affiliation: Department of Mathematics, Swansea University, Swansea SA2 8PP, United Kingdom
Email: z.sobol@swansea.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04453-X
PII: S 0002-9947(08)04453-X
Keywords: Non-divergence semi-linear equation, critical exponents
Received by editor(s): December 20, 2004
Received by editor(s) in revised form: September 15, 2006
Published electronically: September 26, 2008
Additional Notes: The research of the first named author was supported by the Institute of Advanced Studies of the University of Bristol via the Benjamin Meaker Fellowship. The second named author was supported by the Forchheimer Visiting Fellowship, Jerusalem. This research was supported in part by the Volkswagen-Stiftung through the RiP-programme at the Mathematisches Forschungsinstitut Oberwolfach, Germany.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.