Comptes Rendus
Partial Differential Equations
An extreme variation phenomenon for some nonlinear elliptic problems with boundary blow-up
Comptes Rendus. Mathématique, Volume 339 (2004) no. 10, pp. 689-694.

Let Ω be a smooth bounded domain in RN (N2) and Γ be a non-empty open and closed subset of ∂Ω. Denote by B either the Dirichlet or the mixed boundary operator on ΓB:=ΩΓ when ΓΩ. We consider the nonlinear elliptic problem Δu+au=b(x)f(u) in Ω, subject to Bu=0 on ΓB when ΓB, where a is a real number, b is a continuous non-negative function on Ω¯, while f0 is continuous on [0,) such that f(u)/u is increasing on (0,). Assuming that f varies rapidly at infinity with index ∞ (i.e., limuf(λu)/f(u)=λ for all λ>0), we establish the uniqueness of the positive solution satisfying u= on Γ and describe its blow-up rate via the extreme value theory.

Soit Ω un domaine borné, régulier de RN (N2) et Γ un sous-ensemble ouvert et fermé de ∂Ω. On désigne par B ou bien une condition de Dirichlet ou bien une condition mixte sur ΓB:=ΩΓ si ΓΩ. On étudie le problème elliptique non-linéaire Δu+au=b(x)f(u) dans Ω, avec la condition Bu=0 sur ΓB si ΓB, où a est un réel, b est une fonction continue non-négative dans Ω¯ et f0 est continue sur [0,) telle que f(u)/u est strictement croissante sur (0,). Supposons que f varie rapidement à l'infini d'index ∞ (i.e., limuf(λu)/f(u)=λ pour tout λ>0), on établit alors l'unicité de la solution positive avec u= sur Γ et on décrit le taux d'explosion au bord en utilisant la théorie des valeurs extrêmes.

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DOI: 10.1016/j.crma.2004.10.005
Florica-Corina Cîrstea 1

1 School of Computer Science and Mathematics, Victoria University of Technology, PO Box 14428, Melbourne, VIC 8001, Australia
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Florica-Corina Cîrstea. An extreme variation phenomenon for some nonlinear elliptic problems with boundary blow-up. Comptes Rendus. Mathématique, Volume 339 (2004) no. 10, pp. 689-694. doi : 10.1016/j.crma.2004.10.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.10.005/

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

[5] F.-C. Cîrstea; V. Rădulescu 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] F.-C. Cîrstea; V. Rădulescu 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] H. Rademacher Einige besondere Probleme der partiellen Differentialgleichungen, Die Differential und Integralgleichungen der Mechanik und Physik I, Rosenberg, New York, 1943, pp. 838-845

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