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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On stable entire solutions of semi-linear elliptic equations with weights
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by Craig Cowan and Mostafa Fazly PDF
Proc. Amer. Math. Soc. 140 (2012), 2003-2012 Request permission

Abstract:

We are interested in the existence versus non-existence of non-trivial stable sub- and super-solutions of \begin{equation} -\operatorname {div}( \omega _1 \nabla u) = \omega _2 f(u) \qquad \text {in}\ \ \mathbb {R}^N, \end{equation} with positive smooth weights $\omega _1(x),\omega _2(x)$. We consider the cases $f(u) = e^u, u^p$ where $p>1$ and $-u^{-p}$ where $p>0$. We obtain various non-existence results which depend on the dimension $N$ and also on $p$ and the behaviour of $\omega _1,\omega _2$ near infinity. Also the monotonicity of $\omega _1$ is involved in some results. Our methods here are the methods developed by Farina. We examine a specific class of weights $\omega _1(x) = ( |x|^2 +1)^\frac {\alpha }{2}$ and $\omega _2(x) = ( |x|^2+1)^\frac { \beta }{2} g(x)$, where $g(x)$ is a positive function with a finite limit at $\infty$. For this class of weights, non-existence results are optimal. To show the optimality we use various generalized Hardy inequalities.
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Additional Information
  • Craig Cowan
  • Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
  • MR Author ID: 815665
  • Email: ctcowan@stanford.edu
  • Mostafa Fazly
  • Affiliation: Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2
  • MR Author ID: 822619
  • Email: fazly@math.ubc.ca
  • Received by editor(s): February 3, 2011
  • Published electronically: September 30, 2011
  • Additional Notes: This work is supported by a University Graduate Fellowship and is part of the second author’s Ph.D. dissertation in preparation under the supervision of N. Ghoussoub.
  • Communicated by: James E. Colliander
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 140 (2012), 2003-2012
  • MSC (2010): Primary 35B08; Secondary 35J61, 35A01
  • DOI: https://doi.org/10.1090/S0002-9939-2011-11351-0
  • MathSciNet review: 2888188