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

In this paper we study the existence of periodic solutions of the sixth-order equation uvi + Auiv + Bu + u− u3 = 0, where the positive constants A and B satisfy the inequality A2 < 4B. The boundary value problem (P) is considered with the boundary conditions u(0) = u (0) = uiv (0) = 0, u(L) = u (L) = uiv(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 uvi + Auiv + Bu − u+ a(x)u|u|σ = 0, where a is a positive periodic function and σ is a positive constant. The mountain-pass theorem of Brezis–Nirenberg and concentration-compactness arguments are used.  2002 Elsevier Science (USA). All rights reserved.