## 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 $g$.## References

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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