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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
Published electronically: April 19, 2005
MathSciNet review: 2146203
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Abstract: We consider the nonlinear eigenvalue problem

\begin{displaymath}-\Delta u + g(u) = \lambda \sin u \enskip \mbox{in} \enskip ... ..., \enskip u = 0 \enskip \mbox{on} \enskip \partial \Omega, \end{displaymath}

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 $u_\lambda(x) \enskip (x \in \Omega)$ 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

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

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