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

Abstract

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<4 B. 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)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.