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Asymptotic expansion of solutions to nonlinear elliptic eigenvalue problems

Authors:	Tetsutaro Shibata

Institution:	Department of Applied Mathematics, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Japan

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 .

Keywords:	Asymptotic expansion nonlinear elliptic eigenvalue problems

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