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: ON THE EXISTENCE OF POSITIVE SOLUTION FOR AN ELLIPTIC EQUATION OF KIRCHHOFF TYPE VIA MOSER ITERATION METHOD | ON THE EXISTENCE OF POSITIVE SOLUTION FOR AN ELLIPTIC EQUATION OF KIRCHHOFF TYPE VIA MOSER ITERATION METHOD FRANCISCO JULIO S. A. CORREA AND GIOVANY M. FIGUEIREDO Received 18 November 2005 Revised 11 April 2006 Accepted 18 April 2006 Dedicated to our dear friend and collaborator Professor Claudianor O. Alves We investigate the questions of existence of positive solution for the nonlocal problem -M u 2 Au f A u in o and u 0 on do where o is a bounded smooth domain of RN and M and f are continuous functions. Copyright 2006 F. J. S. A. Correa and G. M. Figueiredo. 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 In this paper we study some questions related to the existence of positive solution for the nonlocal elliptic problem - M u 2 Au f A u in o u 0 on do A where o is a bounded smooth domain M R R is a function whose behavior will be stated later f R X R R is a given nonlinear function and II II is the usual norm in Hq o given by Huh2 Vu 2 and finally through this work J u denotes the integral Jo u x dx. The main goal of this paper is to establish conditions on M and f under which problem P a possesses a positive solution. Problem P a is called nonlocal because of the presence of the term M Huh2 which implies that the equation in P a is no longer a pointwise identity. This provokes some mathematical difficulties which make the study of such a problem particular interesting. Hindawi Publishing Corporation Boundary Value Problems Volume 2oO6 Article ID 79679 Pages 1-10 DOI BVP 2006 79679 2 A Kirchhoff-type equation Besides these kinds of problems have motivations in physics. Indeed the operator M u 2 Au appears in the Kirchhoff equation by virtue of this P a is called of the Kirchhoff type which arises in nonlinear vibrations namely utt - M u 2 Au f x u in Q X 0 T u 0 on dQ X 0 T
