The singular extremal solutions of the bi-Laplacian with exponential nonlinearity

Author:
Amir Moradifam

Journal:
Proc. Amer. Math. Soc. **138** (2010), 1287-1293

MSC (2010):
Primary 35J65; Secondary 35J40

DOI:
https://doi.org/10.1090/S0002-9939-09-10257-5

Published electronically:
December 3, 2009

MathSciNet review:
2578522

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Abstract | References | Similar Articles | Additional Information

Abstract: Consider the problem

where is the unit ball in and is a parameter. Unlike the Gelfand problem the natural candidate , for the extremal solution, does not satisfy the boundary conditions, and hence showing the singular nature of the extremal solution in large dimensions close to the critical dimension is challenging. Recently a computer-assisted proof was used to show that the extremal solution is singular in dimensions . Here by an improved Hardy-Rellich inequality we overcome this difficulty and give a simple mathematical proof to show that the extremal solution is singular in dimensions .

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Additional Information

**Amir Moradifam**

Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V6T 1Z2

Email:
a.moradi@math.ubc.ca

DOI:
https://doi.org/10.1090/S0002-9939-09-10257-5

Received by editor(s):
April 23, 2009

Published electronically:
December 3, 2009

Additional Notes:
This work is supported by a Killam Predoctoral Fellowship and is part of the author’s Ph.D. dissertation in preparation under the supervision of N. Ghoussoub.

Communicated by:
Matthew J. Gursky

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.