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Proceedings of the American Mathematical Society

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

 

 

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: Consider the problem

$\displaystyle \left\{ \begin{array}{ll} \Delta^2 u= \lambda e^{u} &\text{in } B,\\ u=\frac{\partial u}{\partial n}=0 &\text{on }\partial B, \end{array} \right.$      

where $ B$ is the unit ball in $ {\mathbb{R}}^N$ and $ \lambda$ is a parameter. Unlike the Gelfand problem the natural candidate $ u=-4\ln(\vert x\vert)$, 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 $ 13\leq N\leq 31$. 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 $ N\geq13$.


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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.