#### Mohsen Timoumi, Infinitely many homoclinic solutions for fourth-order differential equations with locally defined potentials, Vol. 2021 (2021), Article ID 12, pp. 1-10.

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DOI: 10.23952/cot.2021.12

Received April 12, 2021; Accepted October 20, 2021; Published November 1, 2021

Abstract. In this paper, we are concerned with the existence of homoclinic solutions for a class of nonautonomous fourth-order differential equations $u^{(4)}(x)+\omega u''(x)+a(x)u(x)=f(x,u(x)),$  $x\in\mathbb{R}.$ Using a symmetric mountain pass theorem established by Kajikiya, infinitely many homoclinic solutions of the system  are obtained when the function $a$ is only steel bounded from below unnecessary coercive at infinity, and the potential $F(x,u)=\int^{u}_{0}f(x,v)dv$ is only locally defined near the origin with respect to $u$. Our result extends some previously known results in the literature.