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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Asymptotic expansion of solutions to nonlinear elliptic eigenvalue problems
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by Tetsutaro Shibata PDF
Proc. Amer. Math. Soc. 133 (2005), 2597-2604 Request permission

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
  • Received by editor(s): November 7, 2003
  • Published electronically: April 19, 2005
  • Communicated by: David S. Tartakoff
  • © Copyright 2005 American Mathematical Society
  • 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
  • MathSciNet review: 2146203