118 ISSN 1064–5624, Doklady Mathematics, 2009, Vol. 79, No. 1, pp. 118–124. © Pleiades Publishing, Ltd., 2009. Published in Russian in Doklady Akademii Nauk, 2009, Vol. 424, No. 6, pp. 741–747. 1. INTRODUCTION Liouville theorems for positive solutions of nonlin- ear elliptic inequalities have been widely studied in the last years. Besides their intrinsic interest, among other applications, these kind of results are an important tool for proving existence results for related elliptic equa- tions and systems. A systematic study of nonexistence of positive solutions for quasilinear partial differential equations of the form, (1) presents several difficulties. Under special assump- tions, e.g. radial symmetry of the possible solutions, general results have been proved in [10, 11]. Indeed, by considering ground state solutions of (1), that is smooth radial and positive solutions of (1) which satisfies the following decay condition: (2) In [10, 11] it is proved that if A satisfies some structure assumption (see [10, 11] for details), and f : R → R is a given function such that: (i) there exists c > 0 and  > 0 sufficiently small such that for any 0 < t <  we have with q* > 0, then, depending on the operator under con- sideration, it is proved that there exists q* > 0 such that for q ≤ q* problem (1) has no nontrivial ground states. div A x u Du , , ( ) – fu () , x R N , ∈ = u x ( ) x ∞ – → lim 0. = ft () ct q , q 0, > ≥ To be more precise, in the case of A(t) = (1 + t 2 ) –1/2 t or A(t) = t, it is known that while for A(t) = |t| p – 2 t, if p > 1 and N > p, we have Here, without loss of generality (see Remark 2.3 below), we suppose that N > p. Clearly, assumption (i) reduces the nonexistence problem on N ≤ p for (1) to the study of the associated inequality (3) on an exterior domain of R N for solutions u(x) tending to 0 as |x |→ ∞. In the case of the p-Laplacian operator, i.e., A(t) = |t| p – 2 , nonexistence of positive nonradial solutions in an exterior domain has been recently proved in [1, 12], where the model problem f (u) = u q was considered under the assumption, In recent years, this kind of results have been gener- alized in various directions. See [8] for a comprehen- sive study of nonexistence problems and [1, 2, 4–6, 8, 9, 12] for related results. One common feature of most of the recent contributions deals with the case f (u) = u q , q > 0, or nonlinearities which satisfy global conditions (see [12]). The main goal of this paper is to obtain local esti- mates and related Liouville theorems for weak positive solutions of quasilinear elliptic differential inequalities of the form, under weak assumptions on the nonlinearity f. q * N N 2 – ------------ at N 2, > = q * Np 1 – ( ) N p – ---------------------. = div A x u Du , , ( ) – cu q ≥ p 1 – q < q *. ≤ P 1 ( ) div A x u Du , , ( ) – fu () , x ≥ R N ∈ u 0, x R N , ∈ ≥ ⎩ ⎨ ⎧ MATHEMATICS Some Liouville Theorems for Quasilinear Elliptic Inequalities 1 G. Caristi a , E. Mitidieri a , and Corresponding Member of the Russian Academy of Sciences S. I. Pohozaev b Received October 27, 2008 DOI: 10.1134/S1064562409010360 a Dipartimento di Matematica e Informatica, Università degli Studi di Trieste, via A. Valerio, 12/1, I-34127 Trieste b Steklov Mathematical Institute, ul. Gubkina 8, Moscow, 119991, Russia; e-mail: caristi@units.it, mitidier@units.it, pokhozhaev@mi.ras.ru 1 The article was translated by the authors.