## Mohsen Timoumi, Infinitely many homoclinic solutions for fourth-order differential equations with locally defined potentials

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DOI: 10.23952/jnva.3.2019.3.06
Volume 3, Issue 3, 1 December 2019, Pages 305-316

Abstract. In this paper, we are concerned with the existence of infinitely many homoclinic solutions for a class of fourth-order differential equations

$u^{(4)}(x)+\omega u''(x)+a(x)u(x)=f(x,u(x)),$ $\forall x\in\mathbb{R},$

where the function $a\in C(\mathbb{R},\mathbb{R})$ may be negative on a bounded interval and the potential $F(x,u)=\int^{u}_{0}f(x,t)dt$ is only locally defined near the origin with respect to the second variable. Some recent results in the literature are generalized and improved.