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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: Research Article Nontrivial Solutions of the Asymmetric Beam System with Jumping Nonlinear Terms | Hindawi Publishing Corporation Boundary Value Problems Volume 2o10 Article ID 728101 17 pages doi 10.1155 2010 728101 Research Article Nontrivial Solutions of the Asymmetric Beam System with Jumping Nonlinear Terms Tacksun Jung1 and Q-Heung Choi2 1 Department of Mathematics Kunsan National University Kunsan 573-701 Republic of Korea 2 Department of Mathematics Education Inha University Incheon 402-751 Republic of Korea Correspondence should be addressed to Q-Heung Choi qheung@inha.ac.kr Received 8 October 2009 Revised 24 July 2010 Accepted 11 September 2010 Academic Editor Raul F. Manasevich Copyright 2010 T. Jung and Q.-H. Choi. 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 investigate the existence of multiple nontrivial solutions ị y for perturbations b1 u 2 -2 and b2 u 3 - 3 of the beam system with Dirichlet boundary condition Lị b1 ị 3n 2 - 2 in n 2 n 2 X R Ln b2 ị 3n 3 - 3 in -n 2 n 2 X R where u max u 0 and p V are nonzero constants. Here L is the beam operator in R2 and the nonlinearity b1 u 2 - 2 b2 u 3 - 3 crosses the eigenvalues of the beam operator. 1. Introduction Let L be the beam operator in R2 Lu utt uxxxx. In this paper we investigate the existence of multiple nontrivial solutions ị n for perturbations b1 ị 3n 2 -2 and b2 ị 3n 3 -3 of the beam system with Dirichlet boundary condition Lị b1 ị 3n 2 - 2 in f-n n X R s 1 2 27 Ln b2 ị 3n 3 - 3 in f-n n X R 2 2 2 2 7 A n t ịxx n t 0 2 2 1.1 ị x t n ị x f ị -x f n nj nxx n t 0 n n 2 xxy 2 n x t n n x t n -x t 2 Boundary Value Problems where u max u 0 and the nonlinearity b1 u 2 - 2 b2 u 3 - 3 crosses the eigenvalues of the beam operator. This system represents a bending beam supported by cables in the two directions. In 1 2 the authors investigated the multiplicity of solutions of a nonlinear suspension bridge equation in an interval -n 2 n
