An extreme variation phenomenon for some nonlinear elliptic problems with boundary blow-up

Cîrstea, Florica-Corina

doi:10.1016/j.crma.2004.10.005

Partial Differential Equations

An extreme variation phenomenon for some nonlinear elliptic problems with boundary blow-up
[Une phénomène de variation extrême pour quelque problèmes elliptiques non linéaires avec explosion au bord.]

Présenté par : Philippe G. Ciarlet

Cîrstea, Florica-Corina ¹

¹ School of Computer Science and Mathematics, Victoria University of Technology, PO Box 14428, Melbourne, VIC 8001, Australia

Comptes Rendus. Mathématique, Tome 339 (2004) no. 10, pp. 689-694.

Résumé
Abstract

Soit Ω un domaine borné, régulier de $R^{N}$ $(N ⩾ 2)$ et $Γ_{\infty} \neq \emptyset$ un sous-ensemble ouvert et fermé de ∂Ω. On désigne par $B$ ou bien une condition de Dirichlet ou bien une condition mixte sur $Γ_{B} : = \partial Ω ∖ Γ_{\infty}$ si $Γ_{\infty} \neq \partial Ω$ . On étudie le problème elliptique non-linéaire $Δ u + a u = b (x) f (u)$ dans Ω, avec la condition $B u = 0$ sur $Γ_{B}$ si $Γ_{B} \neq \emptyset$ , où a est un réel, b est une fonction continue non-négative dans $\bar{Ω}$ et $f ⩾ 0$ est continue sur $[0, \infty)$ telle que $f (u) / u$ est strictement croissante sur $(0, \infty)$ . Supposons que f varie rapidement à l'infini d'index ∞ (i.e., $\lim_{u \to \infty} f (λ u) / f (u) = λ^{\infty}$ pour tout $λ > 0$ ), on établit alors l'unicité de la solution positive avec $u = \infty$ sur $Γ_{\infty}$ et on décrit le taux d'explosion au bord en utilisant la théorie des valeurs extrêmes.

Let Ω be a smooth bounded domain in $R^{N}$ $(N ⩾ 2)$ and $Γ_{\infty}$ be a non-empty open and closed subset of ∂Ω. Denote by $B$ either the Dirichlet or the mixed boundary operator on $Γ_{B} : = \partial Ω ∖ Γ_{\infty}$ when $Γ_{\infty} \neq \partial Ω$ . We consider the nonlinear elliptic problem $Δ u + a u = b (x) f (u)$ in Ω, subject to $B u = 0$ on $Γ_{B}$ when $Γ_{B} \neq \emptyset$ , where a is a real number, b is a continuous non-negative function on $\bar{Ω}$ , while $f ⩾ 0$ is continuous on $[0, \infty)$ such that $f (u) / u$ is increasing on $(0, \infty)$ . Assuming that f varies rapidly at infinity with index ∞ (i.e., $\lim_{u \to \infty} f (λ u) / f (u) = λ^{\infty}$ for all $λ > 0$ ), we establish the uniqueness of the positive solution satisfying $u = \infty$ on $Γ_{\infty}$ and describe its blow-up rate via the extreme value theory.

Reçu le : 2004-09-16
Accepté le : 2004-09-22
Publié le : 2004-11-05

DOI : 10.1016/j.crma.2004.10.005

Affiliations des auteurs :

Cîrstea, Florica-Corina ¹

¹ School of Computer Science and Mathematics, Victoria University of Technology, PO Box 14428, Melbourne, VIC 8001, Australia

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     title = {An extreme variation phenomenon for some nonlinear elliptic problems with boundary blow-up},
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Cîrstea, Florica-Corina. An extreme variation phenomenon for some nonlinear elliptic problems with boundary blow-up. Comptes Rendus. Mathématique, Tome 339 (2004) no. 10, pp. 689-694. doi : 10.1016/j.crma.2004.10.005. http://www.numdam.org/articles/10.1016/j.crma.2004.10.005/

Bibliographie
Cité par

[1] Bandle, C.; Marcus, M. ‘Large’ solutions of semilinear elliptic equations: Existence, uniqueness, and asymptotic behaviour, J. Anal. Math., Volume 58 (1992), pp. 9-24

[2] Bieberbach, L. $Δ u = e^{u}$ und die automorphen Funktionen, Math. Ann., Volume 77 (1916), pp. 173-212

[3] Bingham, N.H.; Goldie, C.M.; Teugels, J.L. Regular Variation, Cambridge University Press, Cambridge, 1987

[4] F.-C. Cîrstea, Y. Du, General uniqueness results and variation speed for blow-up solutions of elliptic equations, submitted for publication

[5] Cîrstea, F.-C.; Rădulescu, V. Uniqueness of the blow-up boundary solution of logistic equations with absorption, C. R. Acad. Sci. Paris, Ser. I, Volume 335 (2002), pp. 447-452

[6] Cîrstea, F.-C.; Rădulescu, V. Extremal singular solutions for degenerate logistic-type equations in anisotropic media, C. R. Acad. Sci. Paris, Ser. I, Volume 339 (2004), pp. 119-124

[7] Rademacher, H. Einige besondere Probleme der partiellen Differentialgleichungen, Die Differential und Integralgleichungen der Mechanik und Physik I, Rosenberg, New York, 1943, pp. 838-845

[8] Resnick, S.I. Extreme Values, Regular Variation, and Point Processes, Springer-Verlag, New York, 1987

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