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Asymptotic Expansion of Solutions to Nonlinear Elliptic Eigenvalue Problems

Tetsutaro Shibata
Proceedings of the American Mathematical Society
Vol. 133, No. 9 (Sep., 2005), pp. 2597-2604
Stable URL: http://www.jstor.org/stable/4097620
Page Count: 8
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Asymptotic Expansion of Solutions to Nonlinear Elliptic Eigenvalue Problems
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Abstract

We consider the nonlinear eigenvalue problem $-\Delta u + g(u) = \lambda sin u$ in Ω, $u > 0$ in Ω, u = 0 on ∂ Ω, where $\Omega \subset R^N (N \geq 2)$ is an appropriately smooth bounded domain and $\lambda > 0$ is a parameter. It is known that if $\lambda \gg 1$, then the corresponding solution uλ is almost flat and almost equal to π inside Ω. We establish an asymptotic expansion of $u_\lambda(x) (x \in \Omega)$ when $\lambda \gg 1$, which is explicitly represented by g.

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