## The steepest point of the boundary layers of singularly perturbed semilinear elliptic problems

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**356**(2004), 2123-2135 Request permission

## Abstract:

We consider the nonlinear singularly perturbed problem \[ -\epsilon ^2\Delta u = f(u), \quad u > 0 \quad \mbox {in} \quad \Omega , u = 0 \quad \mbox {on} \quad \partial \Omega , \] where $\Omega \subset {\mathbf {R}}^N$ ($N \ge 2$) is an appropriately smooth bounded domain and $\epsilon > 0$ is a small parameter. It is known that under some conditions on $f$, the solution $u_\epsilon$ corresponding to $\epsilon$ develops boundary layers when $\epsilon \to 0$. We determine the steepest point of the boundary layers on the boundary by establishing an asymptotic formula for the slope of the boundary layers with exact second term.## References

- R. A. Adams, Sobolev spaces, Academic Press, New York (1975).
- Henri Berestycki,
*Le nombre de solutions de certains problèmes semi-linéaires elliptiques*, J. Functional Analysis**40**(1981), no. 1, 1–29 (French, with English summary). MR**607588**, DOI 10.1016/0022-1236(81)90069-0 - Philippe Clément and Guido Sweers,
*Existence and multiplicity results for a semilinear elliptic eigenvalue problem*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**14**(1987), no. 1, 97–121. MR**937538** - Robert Stephen Cantrell and Chris Cosner,
*Diffusive logistic equations with indefinite weights: population models in disrupted environments*, Proc. Roy. Soc. Edinburgh Sect. A**112**(1989), no. 3-4, 293–318. MR**1014659**, DOI 10.1017/S030821050001876X - J. M. Fraile, J. López-Gómez, and J. C. Sabina de Lis,
*On the global structure of the set of positive solutions of some semilinear elliptic boundary value problems*, J. Differential Equations**123**(1995), no. 1, 180–212. MR**1359917**, DOI 10.1006/jdeq.1995.1162 - D. G. Figueiredo, On the uniqueness of positive solutions of the Dirichlet problem $-\triangle u = \lambda \sin u$,
*Pitman Res. Notes in Math.***122**(1985), 80–83. - J. García-Melián and J. Sabina de Lis,
*Stationary profiles of degenerate problems when a parameter is large*, Differential Integral Equations**13**(2000), no. 10-12, 1201–1232. MR**1785705** - David Gilbarg and Neil S. Trudinger,
*Elliptic partial differential equations of second order*, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR**737190**, DOI 10.1007/978-3-642-61798-0 - F. A. Howes,
*Singularly perturbed semilinear elliptic boundary value problems*, Comm. Partial Differential Equations**4**(1979), no. 1, 1–39. MR**514718**, DOI 10.1080/03605307908820090 - Wei-Ming Ni and Izumi Takagi,
*Locating the peaks of least-energy solutions to a semilinear Neumann problem*, Duke Math. J.**70**(1993), no. 2, 247–281. MR**1219814**, DOI 10.1215/S0012-7094-93-07004-4 - T. Shibata, Asymptotic formulas for boundary layers and eigencurves for nonlinear elliptic eigenvalue problems, Comm. Partial Differential Equations
**28**(2003), 581–600. - T. Shibata, Three-term asymptotics for the boundary layers of semilinear elliptic eigenvalue problems, to appear in Nonlinear Differential Equations and Applications.

## Additional Information

**T. Shibata**- Affiliation: The Division of Mathematical and Information Sciences, Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima, 739-8521, Japan
- Email: shibata@mis.hiroshima-u.ac.jp
- Received by editor(s): October 3, 2002
- Received by editor(s) in revised form: July 11, 2003
- Published electronically: January 6, 2004
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**356**(2004), 2123-2135 - MSC (2000): Primary 35J65, 35J60
- DOI: https://doi.org/10.1090/S0002-9947-04-03468-3
- MathSciNet review: 2031056