On a Liouville-type comparison principle for solutions of semilinear elliptic partial differential inequalities

Kurta, Vasilii V.

doi:10.1016/j.crma.2005.06.004

Partial Differential Equations

On a Liouville-type comparison principle for solutions of semilinear elliptic partial differential inequalities

Presented by: Pierre-Louis Lions

Kurta, Vasilii V.¹

¹ American Mathematical Society, Mathematical Reviews, 416, Fourth Street, P.O. Box 8604, Ann Arbor, Michigan 48107-8604, USA

Comptes Rendus. Mathématique, Volume 341 (2005) no. 2, pp. 93-96.

Abstract
Résumé

This Note is devoted to the study of a Liouville-type comparison principle for entire weak solutions of semilinear elliptic partial differential inequalities of the form $L u + {| u |}^{q - 1} u ⩽ L v + {| v |}^{q - 1} v$ , where $q > 0$ is a given number and L is a linear (possibly non-uniformly) elliptic partial differential operator of second order in divergent form given formally by the relation

L = \sum_{i, j = 1}^{n} \frac{\partial}{\partial x_{i}} [a_{i j} (x) \frac{\partial}{\partial x_{j}}] .

We assume that

n ⩾ 2

, that the coefficients

a_{i j} (x)

i, j = 1, \dots, n

, are measurable bounded functions on

R^{n}

such that

a_{i j} (x) = a_{j i} (x)

, and that the corresponding quadratic form is non-negative. The results obtained in this work complete similar results on solutions of quasilinear elliptic partial differential inequalities announced in Kurta [C. R. Acad. Sci. Paris, Ser. I 336 (11) (2003) 897–900].

Cette Note est consacré à l'étude d'un principe de comparaison de type Liouville pour des solutions entières faibles d'inégalités aux derivées partielles elliptiques semi-linéaires de la forme $L u + {| u |}^{q - 1} u ⩽ L v + {| v |}^{q - 1} v$ , où $q > 0$ est un nombre donné et L un opérateur aux dérivées partielles (possiblement non-uniformément) elliptique linéaire de deuxième ordre en forme divergente donné formellement par la relation

L = \sum_{i, j = 1}^{n} \frac{\partial}{\partial x_{i}} [a_{i j} (x) \frac{\partial}{\partial x_{j}}] .

Nous supposons que

n ⩾ 2

, que les coefficients

A_{i j} (x)

i, j = 1, \dots, n

, sont des fonctions bornées mesurables en

R^{n}

telles que

a_{i j} (x) = a_{i j} (x)

, et que la forme quadratique correspondante est non-négative. Les résultats obtenus dans ce travail complètent d'autres résultats similaires pour des solutions d'inégalités aux dérivées partielles elliptiques announcés dans Kurta [C. R. Acad. Sci. Paris, Ser. I 336 (11) (2003) 897–900].

Received: 2005-03-07
Accepted: 2005-05-10
Published online: 2005-07-05

DOI: 10.1016/j.crma.2005.06.004

Author's affiliations:

Kurta, Vasilii V. ¹

¹ American Mathematical Society, Mathematical Reviews, 416, Fourth Street, P.O. Box 8604, Ann Arbor, Michigan 48107-8604, USA

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     title = {On a {Liouville-type} comparison principle for solutions of semilinear elliptic partial differential inequalities},
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Kurta, Vasilii V. On a Liouville-type comparison principle for solutions of semilinear elliptic partial differential inequalities. Comptes Rendus. Mathématique, Volume 341 (2005) no. 2, pp. 93-96. doi : 10.1016/j.crma.2005.06.004. http://archive.numdam.org/articles/10.1016/j.crma.2005.06.004/

References
Cited by

[1] Kurta, V.V. On a Liouville-type comparison principle for solutions of quasilinear elliptic inequalities, C. R. Math. Acad. Sci. Paris, Volume 336 (2003) no. 11, pp. 897-900

[2] V.V. Kurta, Some problems of qualitative theory for nonlinear second-order equations, Doctoral Dissert., Steklov Math. Inst., Moscow, 1994

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