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Báo cáo hóa học: " Infinitely many periodic solutions for some second-order differential systems with p(t)Laplacian"

Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí sinh học đề tài : Infinitely many periodic solutions for some second-order differential systems with p(t)Laplacian | Zhang et al. Boundary Value Problems 2011 2011 33 http www.boundaryvalueproblems.eom content 2011 1 33 o Boundary Value Problems a SpringerOpen Journal RESEARCH Open Access Infinitely many periodic solutions for some second-order differential systems with p t -Laplacian Liang Zhang Xian Hua Tang and Jing Chen Correspondence mathspaper@126.com School of Mathematical Sciences and Computing Technology Central South University Changsha Hunan 410083 P. R. China Springer Abstract In this article we investigate the existence of infinitely many periodic solutions for some nonautonomous second-order differential systems with p t -Laplacian. Some multiplicity results are obtained using critical point theory. 2000 Mathematics Subject Classification 34C37 58E05 70H05. Keywords p t -Laplacian Periodic solutions Critical point theory 1. Introduction Consider the second-order differential system with Ji t -Laplacian -d u t p t -2u t u t p t -2u t VF t u t a. e. t e 0 T dt 1.1 _ u 0 - u T u 0 - u T 0 where T 0 F 0 T X RN R and p t e C 0 T R satisfies the following assumptions A ji 0 p T and p 0mt rp t 1 where q 1 which satisfies 1 p- 1 q 1. Moreover we suppose that F 0 T X RN R satisfies the following assumptions A F t x is measurable in t for every x e RN and continuously differentiable in x for a.e. t e 0 T and there exist a e C R R b e L1 0 T R such that F t x a x b t VF t x a x b t for all x e RN and a.e. t e 0 T . The operator u t p t -2u t is said to be p t -Laplacian and becomes Ji-Laplacian dt when p f p a constant . The Ji t -Laplacian possesses more complicated nonlinearity than Ji-Laplacian for example it is inhomogeneous. The study of various mathematical problems with variable exponent growth conditions has received considerable attention in recent years. These problems are interesting in applications and raise many mathematical problems. One of the most studied models leading to problem of this type is the model of motion of electro-rheological fluids which are .