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: ENTIRE POSITIVE SOLUTION TO THE SYSTEM OF NONLINEAR ELLIPTIC EQUATIONS | ENTIRE POSITIVE SOLUTION TO THE SYSTEM OF NONLINEAR ELLIPTIC EQUATIONS LINGYUN QIU AND MIAOXIN YAO Received 8 November 2005 Revised 12 May 2006 Accepted 15 May 2006 The second-order nonlinear elliptic system -Au f1 x ua g1 x u-ỉ hi x uYP v -Av f2 x va g2 x v-ỉ h2 x vYP u with 0 a ỉ Y 1 is considered in RN. Under suitable hypotheses on functions fi gi hi i 1 2 and P it is shown that this system TO x CCPCCPC IP TOTO tl T TO TOTO c itivp c toI inti x TO Í1Í lit cl 2 9 c TO N t cl 2 9 c TO N i TO V ũ 1 di toU l-U 11 possesses an entire positive solution u v e Cloc R S Cloc R 0 9 1 such that both u and v are bounded below and above by positive constant multiples of x12-W for all x 1. Copyright 2006 L. Qiu and M. Yao. 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 This paper is concerned with the second-order nonlinear elliptic system -Au f1 x ua g1 x u-ỉ h1 x uYP v x e RN N 3 -Av f2 x va g2 x v-ỉ h2 x vYP u where A is the Laplacian operator 0 a ỉ Y 1 are constants the functions fi gi hi i 1 2 are nonnegative and locally Holder continuous with exponent 9 e 0 1 in RN and P R R is a continuous differentiable function where R 0 to R 0 ro . We are interested in the study of the existence of entire positive solutions u x v x to which satisfy the condition that each of its elements decays between two positive multiples of x 2-N as x tends to infinity. By an entire solution of is meant a pair of ti 1 TO ntl To TOC til nt r cl 2 9 c TO N XV Cl 2 9 c TO N mil 1 toTi cTOtlctlTOC t 1 It TOt TOIITOm 11 TO 1 TO t 1 1 TO TO N functions u v e Cloc R S Cloc R which satisfies at every point x in R . The existence of entire positive solutions of the equation Au f x u 0 x e Rn N 3 Hindawi Publishing Corporation Boundary Value Problems Volume 2006 Article ID 32492 Pages 1-11 DOI BVP .
