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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: SECOND-ORDER ESTIMATES FOR BOUNDARY BLOWUP SOLUTIONS OF SPECIAL ELLIPTIC EQUATIONS CLAUDIA ANEDDA, ANNA BUTTU, AND GIOVANNI | SECOND-ORDER ESTIMATES FOR BOUNDARY BLOWUP SOLUTIONS OF SPECIAL ELLIPTIC EQUATIONS CLAUDIA ANEDDA ANNA BUTTU AND GIOVANNI PORRU Received 20 October 2005 Accepted 7 November 2005 We find a second-order approximation of the boundary blowup solution of the equation Au eu u 1 with p 0 in a bounded smooth domain o c RN. Furthermore we consider the equation Au eu e . In both cases we underline the effect of the geometry of the domain in the asymptotic expansion of the solutions near the boundary do. Copyright 2006 Claudia Anedda et al. 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 Let o c RN be a bounded smooth domain. In 1916 Bieberbach 10 has investigated the problem Au eu in o u x TO as x do 1.1 and has proved the existence of a classical solution called a boundary blowup explosive large solution. Moreover if s s x denotes the distance from x to do we have 10 u x - log 2 s2 x 0 as x do. Recently Bandle 4 has improved the previous estimate finding the expansion 2 u x log-jAr N - 1 K x S x o 8 x 1.2 02 x where K x denotes the mean curvature of do at the point x nearest to x and o S has the usual meaning. Boundary estimates for various nonlinearities have been discussed in several papers see for example 1 3 5 8 13-16 . In Section 2 of the present paper we investigate boundary blowup solutions of the equation Au eululP 1 with p 0 p 1. We prove the estimate u x O ổ p-1 N - 1 K x ổ o ổ 1-p O 1 ổ o ổ 1-2p 1.3 Hindawi Publishing Corporation Boundary Value Problems Volume 2006 Article ID 45859 Pages 1-12 DOI 10.1155 BVP 2006 45859 2 Second-order estimates where O ổ is defined by the equation 00 I 2F t s -1 2 s t F t LT dT -0 1.4 K x is the mean curvature of the surface x e Q s x constant and 0 1 denotes a bounded quantity. In Section 3 we consider boundary blowup solutions of the equation Aw eu e .
