Ground State Homoclinic Solutions for a Class of Superquadratic Fourth-Order Differential Equations

Abstract

In the present paper, we consider the fourth-order differential equation

$$\begin{aligned} u^{(4)}(x)+\omega u''(x)+a(x)u(x)=f(x,u(x)),\ \forall x\in {\mathbb {R}}\quad \quad \quad \quad (1) \end{aligned}$$

in which $\omega$ represents a constant, $a\in C({\mathbb {R}},{\mathbb {R}})$ and $f\in C({\mathbb {R}}^{2},{\mathbb {R}})$. We are concerned with the existence of ground state homoclinic solution for (1) when a is unnecessary positive and $F(x,u)=\int ^{u}_{0}f(x,t)dt$ satisfies a kind of superquadratic conditions due to Ding and Luan. For the proof, we apply a variant generalized weak linking theorem developed by Schechter and Zou. Some results in the literature are generalized and improved.