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Proceedings of the American Mathematical Society

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

 
 

 

Asymptotic expansion of solutions to nonlinear elliptic eigenvalue problems


Author: Tetsutaro Shibata
Journal: Proc. Amer. Math. Soc. 133 (2005), 2597-2604
MSC (2000): Primary 35J60; Secondary 35P30
DOI: https://doi.org/10.1090/S0002-9939-05-08114-1
Published electronically: April 19, 2005
MathSciNet review: 2146203
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Abstract: We consider the nonlinear eigenvalue problem \[ -\Delta u + g(u) = \lambda \sin u \quad \mathrm {in} \quad \Omega , \quad u > 0\quad \mathrm {in} \quad \Omega , \quad u = 0 \quad \mathrm {on} \quad \partial \Omega , \] where $\Omega \subset {\mathbf {R}}^N\ (N \ge 2)$ is an appropriately smooth bounded domain and $\lambda > 0$ is a parameter. It is known that if $\lambda \gg 1$, then the corresponding solution $u_\lambda$ is almost flat and almost equal to $\pi$ inside $\Omega$. We establish an asymptotic expansion of $u_\lambda (x) \quad (x \in \Omega )$ when $\lambda \gg 1$, which is explicitly represented by $g$.


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Additional Information

Tetsutaro Shibata
Affiliation: Department of Applied Mathematics, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Japan
Email: shibata@amath.hiroshima-u.ac.jp

Keywords: Asymptotic expansion, nonlinear elliptic eigenvalue problems
Received by editor(s): November 7, 2003
Published electronically: April 19, 2005
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2005 American Mathematical Society