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Báo cáo hóa học: "RICCATI INEQUALITY AND OSCILLATION CRITERIA FOR PDE WITH p-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: RICCATI INEQUALITY AND OSCILLATION CRITERIA FOR PDE WITH p-LAPLACIAN | RICCATI INEQUALITY AND OSCILLATION CRITERIA FOR PDE WITH p-LAPLACIAN ZHITING XU Received 1 November 2003 Revised 25 December 2004 Accepted 25 December 2004 Oscillation criteria for PDE with p-Laplacian div A x Du p-2Du p x u p-2u 0 are obtained via Riccati inequality. Some of them are extensions of the results for the second-order linear ODE to this equation. Copyright 2006 Zhiting Xu. 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 are interested in obtaining oscillation criteria for the solutions of the second-order partial differential equation PDE with p-Laplacian div A x Du p-2Du p x u p-2u 0 1.1 in the exterior domain 0 1 x e RN xh 1 where p 1 x x1 . xN e RN N 2 Du du dx1 . du dxN xh is usual Euclidean norm in RN. Throughout this paper we will assume that A1 p e Cpoc 0 1 0 p 1 and p 1 constant A2 a Aij x NxN is a real symmetric positive define matrix function with Aij e C 0 1 i j 1 . N and 0 p 1. Denote by Amin x the smallest eigenvalue of A. We suppose that there is a function A e C 1 to R such that min min x A r for r 1 1.2 x n AH where HAH denotes the norm of the matrix A that is HAH ZNj 1 A2j x 1 2 and q is the conjugate number to p that is q p p - 1 . By a solution of 1.1 we mean a function u x e C2c p 0 1 which satisfies 1.1 almost every on 0 1 . We restrict our attention to the nontrivial solution u x of 1.1 that is to the solution u x such that supxe0 1 u x 0. A nontrivial solution of 1.1 Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006 Article ID 63061 Pages 1-10 DOI 10.1155 JIA 2006 63061 2 Riccati inequality and oscillation criteria is said to be oscillatory if u has zero on O a for every a 1. Equation 1.1 is said to be oscillatory if every solution if any exists is oscillatory. Conversely 1.1 is nonoscillatory if there .