Periodic and Homoclinic Solutions of Extended Fisher–Kolmogorov Equations

In this paper we study the existence of periodic solutions of the fourth-order equations uiv − pu − a x u + b x u3 = 0 and uiv − pu + a x u − b x u3 = 0, where p is a positive constant, and a x and b x are continuous positive 2Lperiodic 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.W e study also the homoclinic solutions for the fourth-order equation uiv + 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 concentrationcompactness arguments are used.