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báo cáo hóa học:" Research Article Positive and Dead-Core Solutions of Two-Point Singular Boundary Value Problems with φ-Laplacian"

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 Positive and Dead-Core Solutions of Two-Point Singular Boundary Value Problems with φ-Laplacian | Hindawi Publishing Corporation Advances in Difference Equations Volume 2010 Article ID 262854 17 pages doi 10.1155 2010 262854 Research Article Positive and Dead-Core Solutions of Two-Point Singular Boundary Value Problems with -Laplacian Svatoslav Stanek Department of Mathematical Analysis Faculty of Science Palacky University Tr. 17. listopadu 12 771 46 Olomouc Czech Republic Correspondence should be addressed to Svatoslav Stanek stanek@inf.upol.cz Received 18 December 2009 Accepted 15 March 2010 Academic Editor Leonid Berezansky Copyright 2010 Svatoslav Stanek. 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. The paper discusses the existence of positive solutions dead-core solutions and pseudo-dead-core solutions of the singular problem f u f Xf t u u u 0 -au 0 A u T Pu 0 Yu fT A. Here X is a positive parameter a 0 A 0 p 0 Y 0 f is singular at u 0 and f may be singular at u 0. 1. Introduction Consider the singular boundary value problem j u tyA Xf t u t uft X 0 1.1 u 0 - au 0 A u T pu 0 fu T A a A 0 p Y 0 1.2 depending on the parameter X. Here Ộ e C R f satisfies the Carathéodory conditions on 0 T xD D 0 1 p a A X R 0 f e Car 0 T xD f is positive limx 0 f t x y TO for a.e. t e 0 T and each y e R 0 and f may be singular at y 0. Throughout the paper AC 0 T denotes the set of absolutely continuous functions on 0 T and xH max x t t e 0 T is the norm in C 0 T . We investigate positive dead-core and pseudo-dead-core solutions of problem 1.1 1.2 . 2 Advances in Difference Equations A function u e C1 0 T is a positive solution of problem 1.1 1.2 if ệ u e AC 0 T u 0 on 0 T u satisfies 1.2 and 1.1 holds for a.e. t e 0 T . We say that u e C1 0 T satisfying 1.2 is a đeađ-core solution of problem 1.1 1.2 if there exist 0 t1 t2 T such that u 0 on t1 t2 u 0 on 0 T t1 t2 ộ u e AC 0 T and 1.1 holds for a.e. t e 0 T t1