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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 RESULTS FOR CLASSES OF p-LAPLACIAN SEMIPOSITONE EQUATIONS | EXISTENCE RESULTS FOR CLASSES OF p-LAPLACIAN SEMIPOSITONE EQUATIONS SHOBHA ORUGANTI AND R. SHIVAJI Received 22 September 2005 Accepted 10 November 2005 We study positive c1 D solutions to classes of boundary value problems of the form -Apu g x u c in D u 0 on dD where Ap denotes the p-Laplacian operator defined by ApZ div Vz p-2Vz p 1 c 0 is a parameter D is a bounded domain in RN N 2 with dD of class c2 and connected if N 1 we assume that D is a bounded open interval andg x 0 c 0 for some x e D semipositone problems . In particular we first study the case when g x u c Af u - c where A 0 is a parameter and f is a c1 0 to function such that f 0 0 f u 0 for 0 u r and f u 0 for u r. We establish positive constants c0 D r and A D r c such that the above equation has a positive solution when c c0 and A A . Next we study the case when g x u c a x up-1 -uY-1 - ch x logistic equation with constant yield harvesting where Y p and a is a C1 iD function that is allowed to be negative near the boundary of D. Here h is a C1 ÍÃ function satisfying h x 0 for x e D h x ị 0 and maxxeDh x 1. We establish a positive constant c1 D a such that the above equation has a positive solution when c c1. Our proofs are based on subsuper solution techniques. Copyright 2006 S. Oruganti and R. Shivaji. 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 We consider weak solutions to classes of boundary value problems of the form -Apu g x u c in D 1.1 u 0 on ỚD where Ap denotes the p-Laplacian operator defined by ApZ div Vz p-2 Vz p 1 c 0 is a parameter D is a bounded domain in RN N 2 with dD of class c2 and connected if N 1 we assume that D is a bounded open interval and g x 0 c 0 for some x e D semipositone problems . By a weak solution to 1.1 we mean a function u e w0 p D Hindawi Publishing Corporation Boundary Value .
