Ž . Journal of Mathematical Analysis and Applications 252, 256277 2000 doi:10.1006jmaa.2000.6996, available online at http:www.idealibrary.com on Some Nonexistence Results for Quasilinear Elliptic Problems Evgeny Galakhov 1 Dipartimento di Scienze Matematiche, Uni  ersita degli Studi di Trieste, ` Piazzale Europa 1, 34100 Trieste, Italy Submitted by William F. Ames Received October 29, 1999 1. INTRODUCTION This paper is devoted to nonexistence of positive solutions for some quasilinear elliptic inequalities and their systems in a bounded domain    N , N  1, without prescribing any boundary conditions. We can formulate the typical problem that we shall study as follows: ‘‘Let  be a second order differential operator in divergence form and let f :        be a given function. What are the sufficient conditions that imply the nonexistence of positive solutions of  u  f x , u in  , 1.1 Ž . Ž . Ž . when u  S, S being a suitable functional class that depends on , f , and ?’’ The origin of the problem dates back to the classical Liouville theorem for the Laplacian. In the case    N , its nonlinear versions have been studied by many authors in connection with associated Dirichlet problems Ž  . in bounded domains see 1 and references therein . Most results in this Ž  . direction deal with the class of radial solutions see 10, 12 . However,  methods developed by Gidas and Spruck for the Laplace equation 6 and by Mitidieri and Pohozaev for a wide class of quasilinear elliptic inequali-   ties 7  9 allow us to obtain sharp nonexistence results without any assumptions concerning the behaviour of eventual solutions. Here we  adapt for our purposes some of the techniques developed in 9 . 1 On leave of absence from Department of Differential Equations, Moscow State Aviation Institute, Volokolamskoe shosse 4, 125871, Moscow, Russia. 256 0022-247X00 $35.00 Copyright  2000 by Academic Press All rights of reproduction in any form reserved.