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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A Neumann problem with critical exponent in nonconvex domains and Lin-Ni's conjecture


Authors: Liping Wang, Juncheng Wei and Shusen Yan
Journal: Trans. Amer. Math. Soc. 362 (2010), 4581-4615
MSC (2010): Primary 35B25, 35J60; Secondary 35B33
Published electronically: April 22, 2010
MathSciNet review: 2645043
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Abstract: We consider the following nonlinear Neumann problem:

$\displaystyle \left\{\begin{array}{lll} -\Delta u + \mu u = u^{\frac{N+2}{N-2}}... ... \frac{\partial u}{\partial n}=0 & \mbox{on} \partial\Omega, \end{array}\right.$

where $ \Omega \subset \mathbb{R}^N$ is a smooth and bounded domain, $ \mu > 0$ and $ n$ denotes the outward unit normal vector of $ \partial\Omega$. Lin and Ni (1986) conjectured that for $ \mu$ small, all solutions are constants. We show that this conjecture is false for all dimensions in some (partially symmetric) nonconvex domains $ \Omega$. Furthermore, we prove that for any fixed $ \mu$, there are infinitely many positive solutions, whose energy can be made arbitrarily large. This seems to be a new phenomenon for elliptic problems in bounded domains.


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Additional Information

Liping Wang
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
Address at time of publication: Department of Mathematics, East China Normal University, 500 Dong Chuan Road, Shanghai, China
Email: lpwang@math.ecnu.edu.cn

Juncheng Wei
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
Email: wei@math.cuhk.edu.hk

Shusen Yan
Affiliation: School of Mathematics, Statistics and Computer Science, The University of New England, Armidale, NSW 2351, Australia
Email: syan@turing.une.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9947-10-04955-X
PII: S 0002-9947(10)04955-X
Received by editor(s): May 23, 2008
Published electronically: April 22, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.