Periodic and homoclinic solutions of some semilinear sixth-order differential equations

摘要

In this paper we study the existence of periodic solutions of the sixth-order equation u(vi) + Au-iv + Bu" + u - u(3) = 0, where the positive constants A and B satisfy the inequality A(2) < 4B. The boundary value problem (P) is considered with the boundary conditions u(0) = u"(0) = u(iv)(0) = 0, u(L) = u"(L) = u(iv) (L) = 0. Existence of nontrivial solutions for (P) is proved using a minimization theorem and a multiplicity result using Clark's theorem. We study also the homoclinic solutions for the sixth-order equation u(vi) + Au-iv + Bu" - u + a(x)uu(sigma) = 0, where a is a positive periodic function and a is a positive constant. The mountain-pass theorem of Brezis-Nirenberg and concentration-compactness arguments are used. (C) 2002 Elsevier Science (USA). All rights reserved. [References: 12]