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Periodic and homoclinic solutions of a class of fourth order equations. (English) Zbl 1235.34066
This paper studies a periodic boundary-value problem for a class of fourth-order nonlinear ordinary differential equations, which includes the stationary solutions of the extended Fisher-Kolmogorov equation. Critical point theory is applied to functionals whose critical points correspond to solutions of the differential equation, and under some conditions on the growth of the nonlinearity, its symmetry and its interaction with the spectrum of the linear part. The existence of one, several, or infinitely many solutions is proved. Also, an existence theorem for homoclinic solutions is established.
MSC:
34B15Nonlinear boundary value problems for ODE
34C25Periodic solutions of ODE
34C37Homoclinic and heteroclinic solutions of ODE
58E50Applications of variational methods in infinite-dimensional spaces