Existence and nonexistence of nontrivial solutions of semilinear sixth-order ordinary differential equations

K e y w o r d s E x t e n d e d Fisher-Kolmogorov equation, Sixth-order equations, Eigenvalue problems~ Clark's theorem, Variational methods. 1. I N T R O D U C T I O N Bistable systems play an important role in the study of spatial patterns. Recently, interest has turned to higher-order model equations involving bistable dynamics, such as the extended Fisher-Kolmogorov (EFK) equation Ou 0% 02u at ~ j ~ 4 + ~ + ~ ~3, ~ > o, (1.1) proposed by Coullet, Elphick and Repanx [1] and Dee and van Saarloos [2]. The EFK equation has appeared in studies of phase transitions, for instance, near a Lifshitz point (cf. [3]). This work was partially supported by Grant MM 904/99 with the Bulgarian Research Foundation and Grant 2002-PF-03 with the University of Rousse. 0893-9659/04/$ see front matter (~) 2004 Elsevier Ltd. All rights reserved. Typeset by J t ~ T E X dei:10. ]016/j.aml.2003.05.014 1208 J.V. CHAPAROVA et al. Whereas the EFK equation anc~ the related Swift-Hohenberg equation (cf. [4-6]) include fourthorder spatial derivatives, certain:phase field models lead to equations involving even higher-order gradients. We mention the equation Ou 0 % . 0 % O~'u u 3 (1.2) Ot Ox ~ + A'o-fix4 ÷ H-d-fix~ ÷ u studied by Gardner and Jones [7] and Caginalp and Fife [8]. Here A and B are constants. In this paper, we study the sixth-order equation (I) u vi + Au i~ + B u '~ ÷ Cu b (x) u 3 = 0, (1.3) which results from equation (1.2) when we seek stationary solutions. Here A and B are arbitrary constants, while C is assumed to be a positive constant. We assume that b(x) is an even continuous positive 2L-periodic function. We denote by (P) the boundary value problem for equation (I) in the interval (0, L) with the boundary conditions u 0, u/~-0, u ~v = 0, at x = 0, L. We shall prove the existence of 2L-periodic solutions of equation (I), which are antisymmetric with respect to x = 0 and x = L, by extending the solutions u of Problem (P) on [0, L] as odd functions on I -L , L] and then continuing the resulting function periodically over the real line. Problem (P) was studied in [9] under the assumptions that A > 0, A s < 4B, and C = 1. The weak solutions of Problem (P) can be found as critical points of the functional