tailieunhanh - Báo cáo hóa học: " Research Article Existence and Multiplicity Results for Degenerate Elliptic Equations with Dependence on the Gradient"

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 Existence and Multiplicity Results for Degenerate Elliptic Equations with Dependence on the Gradient | Hindawi Publishing Corporation Boundary Value Problems Volume 2007 Article ID 47218 12 pages doi 2007 47218 Research Article Existence and Multiplicity Results for Degenerate Elliptic Equations with Dependence on the Gradient Leonelo Iturriaga and Sebastian Lorca Received 17 October 2006 Revised 2 January 2007 Accepted 9 February 2007 Recommended by Shujie Li We study the existence of positive solutions for a class of degenerate nonlinear elliptic equations with gradient dependence. For this purpose we combine a blowup argument the strong maximum principle and Liouville-type theorems to obtain a priori estimates. Copyright 2007 L. Iturriaga and S. Lorca. 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 the following nonvariational problem - Amu f x u Vu - a x g u Vu T in o u 0 on do P T where o is abounded domain with smooth boundary of RN N 3. Am denotes the usual m-Laplacian operators 1 m N and T 0. We will obtain a priori estimate to positive solutions of problem P T under certain conditions on the functions f g a. This result implies nonexistence of positive solutions to T large enough. Also we are interested in the existence of a positive solutions to problem P 0 which does not have a clear variational structure. To avoid this difficulty we make use of the blow-up method over the solutions to problem P T which have been employed very often to obtain a priori estimates see . 1 2 . This analysis allows us to apply a result due to 3 which is a variant of a Rabinowitz bifurcation result. Using this result we obtain the existence of positive solutions. Throughout our work we will assume that the nonlinearities f and g satisfy the following conditions. H1 f o X R X RN R is a nonnegative continuous function. H2 g R X Rn R is a nonnegative continuous function. 2 Boundary