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 Global Structure of Nodal Solutions for Second-Order m-Point Boundary Value Problems with Superlinear Nonlinearities | Hindawi Publishing Corporation Boundary Value Problems Volume 2011 Article ID 715836 12 pages doi 2011 715836 Research Article Global Structure of Nodal Solutions for Second-Order m-Point Boundary Value Problems with Superlinear Nonlinearities Yulian An Department of Mathematics Shanghai Institute of Technology Shanghai 200235 China Correspondence should be addressed to Yulian An anyulian@ Received 8 May 2010 Revised 1 August 2010 Accepted 23 September 2010 Academic Editor Feliz Manuel Minhos Copyright 2011 Yulian An. 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. We consider the nonlinear eigenvalue problems u Xf u 0 0 t 1 u 0 0 u 1 m-2aiu nf where m 3 Pi e 0 1 and ai 0 for i 1 . m - 2 with 2ml ai 1 and f e C1 R 0 R n C R R satisfies f s s 0 for s f 0 and f0 TO where f0 lim s 0f s s. We investigate the global structure of nodal solutions by using the Rabinowitz s global bifurcation theorem. 1. Introduction We study the global structure of nodal solutions of the problem u Xf u 0 t e 0 1 1-1 m-2 u 0 0 u 1 ai ni . i 1 Here m 3 n e 0 1 and ai 0 for i 1 . m - 2 with 12ai 1 X is a positive parameter and f e C1 R 0 R n C R R . In the case that f0 e 0 to the global structure of nodal solutions of nonlinear second-order m-point eigenvalue problems have been extensively studied see 1-5 and the references therein. However relatively little is known about the global structure of solutions in the case that f0 TO and few global results were found in the available literature when f0 TO fTO. The likely reason is that the global bifurcation techniques cannot be 2 Boundary Value Problems used directly in the case. On the other hand when m-point boundary value condition is concerned the discussion is more difficult since the problem is nonsymmetric and the corresponding operator is .
