Periodic and Homoclinic Solutions of Extended Fisher-Kolmogorov Equations
نویسندگان
چکیده
In this paper we study the existence of periodic solutions of the fourth-order equations u − pu′′ − a x u + b x u3 = 0 and u − 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. We study also the homoclinic solutions for the fourth-order equation u + 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. © 2001 Academic Press
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