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: THE EXACT ASYMPTOTIC BEHAVIOUR OF THE UNIQUE SOLUTION TO A SINGULAR DIRICHLET PROBLE | THE EXACT ASYMPTOTIC BEHAVIOUR OF THE UNIQUE SOLUTION TO A SINGULAR DIRICHLET PROBLEM ZHIJUN ZHANG AND JIANNING YU Received 23 August 2005 Revised 10 November 2005 Accepted 13 November 2005 By Karamata regular variation theory we show the existence and exact asymptotic behaviour of the unique classical solution u e C2 a u n C u near the boundary to a singular Dirichlet problem - Au g u - k x u 0 x e u u an 0 where u is abounded domain with smooth boundary in RN g e C1 0 oo 0 oo lim 0- g t g t -Y for each 0 and some Y 1 and k e Cl c u for some a e 0 1 which is nonnegative on u and may be unbounded or singular on the boundary. Copyright 2006 Z. Zhang and J. Yu. 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 and the main results The purpose of this paper is to investigate the existence and exact asymptotic behaviour of the unique classical solution near the boundary to the following model problem -Au g u - k x u 0 x e u u an 0 where u is a bounded domain with smooth boundary in RN N 1 k e Cl c U for some a e 0 1 which is nonnegative on u and g satisfies g1 g e C1 0 oo 0 oo g s 0 for all s 0 lims 0 g s o. The problem arises in the study of non-Newtonian fluids boundary layer phenomena for viscous fluids chemical heterogeneous catalysts as well as in the theory of heat conduction in electrical conductive materials see 4 7 12 14 . The main feature of this paper is the presence of the two terms the singular term g u which is regular varying at zero of index -y with Y 1 and includes a large class of singular functions and the nonhomogeneous term k x which may be singular on the boundary. This type of nonlinear terms arises in the papers of Diaz and Letelier 6 Lasry and Lions 10 for boundary blow-up elliptic problems. Hindawi Publishing Corporation Boundary Value Problems Volume 2006 Article .
