Comptes Rendus
Partial Differential Equations
Extremal singular solutions for degenerate logistic-type equations in anisotropic media
Comptes Rendus. Mathématique, Volume 339 (2004) no. 2, pp. 119-124.

Let Ω be a smooth bounded domain in RN. Let b0, b≢0 be a continuous function on Ω¯ and consider a closed subset D0 of [b=0]. We study the logistic problem Δu+au=b(x)f(u) in ΩD0, Bu=0 on ∂Ω, and u=+ on D0, where a is a real number, B denotes either the Dirichlet or the mixed boundary operator, and f0 is a smooth function such that f(u)/u is increasing on (0,). In this Note we establish the existence of extremal singular solutions to the above problem, a uniqueness result, and we describe the blow-up at the boundary.

Soit Ω un domaine borné et régulier de RN. Soit b0, b≢0 une fonction continue dans Ω¯ et D0 un sous-ensemble fermé de [b=0]. On étudie le problème logistique Δu+au=b(x)f(u) dans ΩD0, Bu=0 sur ∂Ω, et u=+ sur D0, où a est un réel, B désigne ou bien une condition de Dirichlet ou bien une condition mixte sur ∂Ω, et f0 est une fonction régulière telle que l'application f(u)/u soit croissante sur (0,). Dans cette Note on établit l'existence des solutions singulières extremales, un résultat d'unicité et on décrit également la vitesse d'explosion au bord.

Received:
Published online:
DOI: 10.1016/j.crma.2004.04.025
Florica-Corina Cîrstea 1; Vicenţiu Rădulescu 2

1 School of Computer Science and Mathematics, Victoria University of Technology, PO Box 14428, Melbourne City MC, Victoria 8001, Australia
2 University of Craiova, Department of Mathematics, 13 A. I. Cuza Street, 200585 Craiova, Romania
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Florica-Corina Cîrstea; Vicenţiu Rădulescu. Extremal singular solutions for degenerate logistic-type equations in anisotropic media. Comptes Rendus. Mathématique, Volume 339 (2004) no. 2, pp. 119-124. doi : 10.1016/j.crma.2004.04.025. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.04.025/

[1] F.-C. Cîrstea; V. Rădulescu Existence and uniqueness of blow-up solutions for a class of logistic equations, Commun. Contemp. Math., Volume 4 (2002), pp. 559-586

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

[3] F.-C. Cîrstea, V. Rădulescu, Nonlinear problems with singular boundary conditions arising in population dynamics: a Karamata regular variation theory approach, submitted for publication

[4] Y. Du; Q. Huang Blow-up solutions for a class of semilinear elliptic and parabolic equations, SIAM J. Math. Anal., Volume 31 (1999), pp. 1-18

[5] M. Marcus; L. Véron Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 14 (1997), pp. 237-274

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

[7] E. Seneta Regularly Varying Functions, Lecture Notes in Math., vol. 508, Springer-Verlag, Berlin, 1976

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