Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Existence and Nonexistence Results for a Class of Quasilinear Elliptic Systems | Hindawi Publishing Corporation Boundary Value Problems Volume 2007 Article ID 85621 5 pages doi 2007 85621 Research Article Existence and Nonexistence Results for a Class of Quasilinear Elliptic Systems Said El Manouni and Kanishka Perera Received 18 June 2007 Accepted 20 August 2007 Recommended by Donal O Regan Using variational methods we prove the existence and nonexistence of positive solutions for a class of p -Laplacian systems with a parameter. Copyright 2007 S. El Manouni and K. Perera. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction In a recent paper Perera 1 studied the existence multiplicity and nonexistence of positive classical solutions of the p-Laplacian problem -ApU Af x u in o u 0 on do . where o is a smooth bounded domain in R n 1 ApU div Vu p-2Vu is the p-Laplacian of u 1 p 00 A 0isa parameter and f is a Caratheodory function on o X 0 o satisfying I f x t I Ctp-1 V x t where C denotes a generic positive constant. Assuming f1 3Ổ 0 such that F x t J0 f x T dr 0 when t Ô f2 3t0 0 such that F x t0 0 f3 limsup t o F x t tp 0 uniformly in x and using variational methods the author proved that there are A A such that has no positive solution for A A and at least two positive solutions u1 u2 for A A. A similar result for the semilinear case p 2 was proved by Maya and Shivaji 2 . 2 Boundary Value Problems In the present paper we consider the corresponding p q -Laplacian system -ApU AFu x u v in Q -Aqv AFv x u v in Q u v 0 on dQ where 1 p q 00 and F is a C1-function on Q X 0 o X 0 to satisfying Ft x t s I Ctasp 1 Fs x t s I Cta 1sP V x t s for some a p 0 with a 1 p p 1 q 1. We will extend the results of Perera 1 to this system as follows. Theorem . Thereisa A such that has no positive solution for A A. Theorem . Assume F1 38 0 such that F x t s 0
