Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Fourth order elliptic system with dirichlet boundary condition | Jung and Choi Journal of Inequalities and Applications 2011 2011 60 http content 2011 1 60 RESEARCH Journal of Inequalities and Applications a SpringerOpen Journal Open Access Fourth order elliptic system with dirichlet boundary condition Tacksun Jung1 and Q-Heung Choi2 Correspondence tsjung@kunsan. department of Mathematics Kunsan National University Kunsan 573-701 Korea Full list of author information is available at the end of the article Abstract We investigate the multiplicity of the solutions of the fourth order elliptic system with Dirichlet boundary condition. We get two theorems. One theorem is that the fourth order elliptic system has at least two nontrivial solutions when 1k c 1k 1 and lk n 1k n - c a b 1k n 1 1k n 1 - c . We prove this result by the critical point theory and the variation of linking method. The other theorem is that the system has a unique nontrivial solution when 1k c 1k 1 and 1k 1k - c 0 a b 1k 1 1k 1 - c . We prove this result by the contraction mapping principle on the Banach space. AMS Mathematics Subject Classification 35J30 35J48 35J50 Keywords Fourth order elliptic system fourth order elliptic equation variation linking theorem contraction mapping principle 1. Introduction Let o be a smooth bounded region in Rn with smooth boundary 90. Let 11 12 . 1k . be the eigenvalues of -A with Dirichlet boundary condition in o. In this paper we investigate the multiplicity of the solutions of the following fourth order elliptic system with Dirichlet boundary condition A2u cAu a u v 1 1 in o A2v cAv b u v 1 1 in o u 0 v 0 Au 0 Av 0 on 30 where c e R u max u 0 and a b e R are constant. The single fourth order elliptic equations with nonlinearities of this type arises in the study of travelling waves in a suspension bridge 6 or the study of the static deflection of an elastic plate in a fluid and have been studied in the context of the second order elliptic operators. In particular Lazer .
