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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: EXISTENCE AND NONEXISTENCE OF POSITIVE SOLUTIONS FOR QUASILINEAR SYSTEMS | EXISTENCE AND NONEXISTENCE OF POSITIVE SOLUTIONS FOR QUASILINEAR SYSTEMS HAIYAN WANG Received 14 October 2005 Revised 6 February 2006 Accepted 14 February 2006 The paper deals with the existence and nonexistence of positive solutions for a class of p-Laplacian systems. We investigate the effect of the size of the domain on the existence of positive solution for the problem in sublinear cases. We will use fixed point theorems in a cone. Copyright 2006 Haiyan Wang. 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 this paper we consider the existence and nonexistence of positive solutions to the boundary value problem of the p-Laplacian system tN 1 úị t p 2u ị t tN 1 fi u1 . Un 0 0 t R i 1 . n úị 0 ui R 0 i 1 . n 1.1 where p 1 N 1 R 0 and fi is nonnegative continuous i 1 . n. Such a problem arises when we seek the radial solutions of the following elliptic system ApUi f u1 . un in B i 1 . n ui 0 on dQ i 1 . n 1.2 where ApUi div ui p-2 Ui i 1 . n p 1 B x e RN x R R 0. Equation 1.2 covers several important cases. When p 2 1.2 becomes the elliptic system Aui fi u1 . un in B i 1 . n ui 0 on dB i 1 . n. 1.3 Hindawi Publishing Corporation Boundary Value Problems Volume 2006 Article ID 71534 Pages 1-9 DOI 10.1155 BVP 2006 71534 2 Existence and nonexistence of positive solutions When n 1 1.2 becomes the usual p-Laplacian -Apu f u in B _n 1.4 u 0 on ỚB. When n 1 and p 2 1.2 becomes the usual Laplacian -Au f u in B 1.5 u 0 on ỚB. In several papers 6 8 Wang studied the existence of nontrivial solutions of 1.1 for a fixed R 0. It was shown that 1.1 for a fixed R 0 has a nontrivial solution for sublinear nonlinearities. Related results can also be found in 1 . In this paper we investigate the effect of the size of the domain on the existence and nonexistence of positive solutions of the quasilinear .
