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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On stable entire solutions of semi-linear elliptic equations with weights


Authors: Craig Cowan and Mostafa Fazly
Journal: Proc. Amer. Math. Soc. 140 (2012), 2003-2012
MSC (2010): Primary 35B08; Secondary 35J61, 35A01
Published electronically: September 30, 2011
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Abstract: We are interested in the existence versus non-existence of non-trivial stable sub- and super-solutions of

$\displaystyle -\operatorname {div}( \omega _1 \nabla u) = \omega _2 f(u)$% latex2html id marker 437 $\displaystyle \qquad \text {in}\ \ \mathbb{R}^N,$ (1)

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) = ( \vert x\vert^2 +1)^\frac {\alpha }{2}$ and $ \omega _2(x) = ( \vert x\vert^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
Email: ctcowan@stanford.edu

Mostafa Fazly
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2
Email: fazly@math.ubc.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11351-0
PII: S 0002-9939(2011)11351-0
Keywords: Semi-linear elliptic equations, Hardy’s inequality, stable solutions
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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.