Periodic and Homoclinic Solutions of Extended Fisher–Kolmogorov Equations

In this paper we study the existence of periodic solutions of the fourth-order equations u^iv − pu″ − a(x)u + b(x)u³ = 0 and u^iv − pu″ + a(x)u − b(x)u³ = 0, where p is a positive constant, and a(x) and b(x) are continuous positive 2L-periodic functions. The boundary value problems (P₁) and (P₂) for these equations are considered respectively with the boundary conditions u(0) = u(L) = u″(0) = u″(L) = 0. Existence of nontrivial solutions for (P₁) is proved using a minimization theorem and a multiplicity result using Clark's theorem. Existence of nontrivial solutions for (P₂) is proved using the symmetric mountain-pass theorem. We study also the homoclinic solutions for the fourth-order equation u^iv + pu″ + a(x)u − b(x)u² − c(x)u³ = 0, where p is a constant, and a(x), b(x), and c(x) are periodic functions. The mountain-pass theorem of Brezis and Nirenberg and concentration-compactness arguments are used.