Infinitely many solutions of systems of Kirchhoff-type equations with general potentials

Abstract

This paper is concerned with the following systems of Kirchhoff-type equations: \begin{equation} \begin{cases} -\left (a+b\int _{\mathbb {R}^{N}}|\nabla u|^{2}\mathrm {d}x\right )\Delta u\\ \quad +V(x)u=F_{u}(x, u, v)\quad & x\in \mathbb {R}^{N},\\ -\left (c+d\int _{\mathbb {R}^{N}}|\nabla v|^{2}\mathrm {d}x\right )\Delta v\\ \quad +V(x)v=F_{v}(x, u, v) & x\in \mathbb {R}^{N},\\ u(x)\rightarrow 0,\quad v(x)\rightarrow 0 &\mbox {as } |x|\rightarrow \infty . \end{cases} \end{equation} Under some more relaxed assumptions on $V(x)$ and $F(x, u, v)$, we prove the existence of infinitely many negative-energy solutions for the above system via the genus properties in critical point theory. Some recent results from the literature are greatly improved and extended.