Positive super-solutions to semi-linear second-order non-divergence type elliptic equations in exterior domains
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- by Vladimir Kondratiev, Vitali Liskevich and Zeev Sobol PDF
- Trans. Amer. Math. Soc. 361 (2009), 697-713 Request permission
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.References
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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
- 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.
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 697-713
- MSC (2000): Primary 35J60, 35B33; Secondary 35B05
- DOI: https://doi.org/10.1090/S0002-9947-08-04453-X
- MathSciNet review: 2452821