Partial Differential Equations
On a Liouville-type comparison principle for solutions of quasilinear elliptic inequalities
Comptes Rendus. Mathématique, Volume 336 (2003) no. 11, pp. 897-900.

We characterize in terms of monotonicity basic properties of quasilinear elliptic partial differential operators which make it possible to obtain a Liouville-type comparison principle for entire solutions of quasilinear elliptic partial differential inequalities of the form A(u)+|u|q−1uA(v)+|v|q−1v, which belong only locally to the corresponding Sobolev spaces on n ,n2. We establish that such properties are inherent for a wide class of quasilinear elliptic partial differential operators. Typical examples of such operators are the p-Laplacian and its well-known modifications for 1<p⩽2.

On caractérise en terme de monotonie, des propriétés fondamentales d'opérateurs aux dérivées partielles, elliptiques, quasi-linéaires permettant d'établir un principe de comparaison de type Liouville, des solutions faibles d'inégalités aux dérivée partielles, elliptiques, quasi-linéaires de la forme A(u)+|u|q−1uA(v)+|v|q−1v. Ces solutions appartiennent seulement localement aux espaces de Sobolev correspondant dans n ,n2. On montre que ces propriétés sont valables pour une large classe d'opérateurs aux dérivées partielles elliptiques, quasi-linéaires. Des exemples typiques de tels opérateurs sont le p-laplacien et ses modifications bien connues pour 1<p⩽2.

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DOI: 10.1016/S1631-073X(03)00225-5
Kurta, Vasilii V. 1

1 Mathematical Reviews, 416, Fourth Street, PO Box 8604, Ann Arbor, MI 48107-8604, USA
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Kurta, Vasilii V. On a Liouville-type comparison principle for solutions of quasilinear elliptic inequalities. Comptes Rendus. Mathématique, Volume 336 (2003) no. 11, pp. 897-900. doi : 10.1016/S1631-073X(03)00225-5. http://www.numdam.org/articles/10.1016/S1631-073X(03)00225-5/

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This work was reported by the author at the 981st AMS Meeting in October, 2002.