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

 

Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity


Author: Jaeyoung Byeon
Journal: Trans. Amer. Math. Soc. 362 (2010), 1981-2001
MSC (2000): Primary 35J65, 35J20
Published electronically: November 16, 2009
MathSciNet review: 2574884
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Abstract: Let $ \Omega$ be a bounded domain in $ \mathbf{R}^n,$ $ n \ge 3,$ with a boundary $ \partial \Omega \in C^2.$ We consider the following singularly perturbed nonlinear elliptic problem on $ \Omega$:

$\displaystyle \varepsilon^2 \Delta u - u + f(u) = 0, \ u > 0 \textrm{ on }\Omega, \quad u = 0 \textrm{ on } \partial \Omega, $

where the nonlinearity $ f$ is of subcritical growth. Under rather strong conditions on $ f,$ it has been known that for small $ \varepsilon > 0,$ there exists a mountain pass solution $ u_\varepsilon$ of above problem which exhibits a spike layer near a maximum point of the distance function $ d$ from $ \partial \Omega$ as $ \varepsilon \to 0.$ In this paper, we construct a solution $ u_\varepsilon$ of above problem which exhibits a spike layer near a maximum point of the distance function under certain conditions on $ f$, which we believe to be almost optimal.


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Additional Information

Jaeyoung Byeon
Affiliation: Department of Mathematics, Pohang University of Science and Technology, Pohang, Kyungbuk 790-784, Republic of Korea
Email: jbyeon@postech.ac.kr

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04746-1
PII: S 0002-9947(09)04746-1
Received by editor(s): October 4, 2006
Received by editor(s) in revised form: December 12, 2007
Published electronically: November 16, 2009
Additional Notes: This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2007-313-C00047)
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.