Communications of the
Korean Mathematical Society

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024



Commun. Korean Math. Soc. 2022; 37(1): 137-161

Online first article January 3, 2022      Printed January 31, 2022

Copyright © The Korean Mathematical Society.

Infinitely many homoclinic solutions for different classes of fourth-order differential equations

Mohsen Timoumi

Faculty of Sciences of Monastir


In this article, we study the existence and multiplicity of homoclinic solutions for the following fourth-order differential equation $$u^{(4)}(x)+omega u''(x)+a(x)u(x)=f(x,u(x)), forall xinmathbb{R} leqno(1)$$ where $a(x)$ is not required to be either positive or coercive, and $F(x,u)=int^{u}_{0}f(x,v)dv$ is of subquadratic or superquadratic growth as $left|uight|ightarrowinfty$, or satisfies only local conditions near the origin (i.e., it can be subquadratic, superquadratic or asymptotically quadratic as $|u|ightarrowinfty$). To the best of our knowledge, there is no result published concerning the existence and multiplicity of homoclinic solutions for (1) with our conditions. The proof is based on variational methods and critical point theory.

Keywords: Fourth-order differential equations, homoclinic solutions, critical points, subquadratic growth, superquadratic growth, local conditions

MSC numbers: Primary 34C37, 58E05, 70H05