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

In the present paper, we consider the fourth-order differential equation u(4)(x)+ωu′′(x)+a(x)u(x)=f(x,u(x)),∀x∈R(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u^{(4)}(x)+\omega u''(x)+a(x)u(x)=f(x,u(x)),\ \forall x\in {\mathbb {R}}\quad \quad \quad \quad (1) \end{aligned}$$\end{document}in which ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document} represents a constant, a∈C(R,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\in C({\mathbb {R}},{\mathbb {R}})$$\end{document} and f∈C(R2,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in C({\mathbb {R}}^{2},{\mathbb {R}})$$\end{document}. We are concerned with the existence of ground state homoclinic solution for (1) when a is unnecessary positive and F(x,u)=∫0uf(x,t)dt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(x,u)=\int ^{u}_{0}f(x,t)dt$$\end{document} 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.