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

MathSciNet review:
2888188

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Abstract: We are interested in the existence versus non-existence of non-trivial stable sub- and super-solutions of

with positive smooth weights . We consider the cases where and where . We obtain various non-existence results which depend on the dimension and also on and the behaviour of near infinity. Also the monotonicity of is involved in some results. Our methods here are the methods developed by Farina. We examine a specific class of weights and , where is a positive function with a finite limit at . 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

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.