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ISSN 1088-6850(e) ISSN 0002-9947(p)

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

Author(s): Liping Wang; Juncheng Wei; Shusen Yan
Journal: Trans. Amer. Math. Soc. 362 (2010), 4581-4615.
MSC (2010): Primary 35B25, 35J60; Secondary 35B33
Posted: April 22, 2010
MathSciNet review: 2645043
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

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

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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