This paper shows the existence of infinitely many solutions for a class of two-point boundary value Kirchhoff-type systems \[ \begin{aligned} & -K_{i}\left(\int^{a}_b|u'_{i}|^{2}dx\right)u''_{i}=\lambda F_{u_{i}}(x,u_{1},\dots,u_{n})+\mu G_{u_{i}}(x,u_{1},\dots,u_{n}),\\ & u_{i}(a)=u_{i}(b)=0,\quad i=1,\dots,n, \end{aligned}\tag{0.1} \] by using Ricceri’s variational principle. The main result is the following theorem.
Theorem 2.1. Assume that there exist positive constants \(\alpha\) and \(\beta\) with \(\beta+\alpha<b-a\) such that \[ F(x,t)\geq0 \text{ for each } (x,t)\in([a,a+\alpha]\cup[b-\beta,b])\times\mathbb R^{n};\tag{\(A_1\)} \]
\[ \liminf_{\xi\rightarrow+\infty}\frac{\int^{b}_{a}\sup_{t\in Q(\xi)}F(x,t)dx}{\xi^{2}}<\frac{4\underline{m}}{n^{2}(b-a)}\limsup_{t\rightarrow+\infty}\frac{\int^{b-\beta}_{a+\alpha}F(x,t)dx}{\sum^{n}_{i=1}\tilde{K_{i}}\left(\frac{\alpha+\beta}{\alpha\beta}t_{i}^{2}\right)} \tag{\(A_2\)} \] (note \(t\rightarrow+\infty\) means \((t_{1},\dots,t_{n})\rightarrow(+\infty,\dots,+\infty))\). Then for each \(\lambda\in(\lambda_{1},\lambda_{2})\), where \[ \begin{aligned} &\lambda_{1}:=\frac{1}{2\limsup_{t\rightarrow+\infty}\frac{\int^{b-\beta}_{a+\alpha}F(x,t)dx}{\sum^{n}_{i=1}\tilde{K_{i}}\left(\frac{\alpha+\beta}{\alpha\beta}t_{i}^{2}\right)}},\\ &\lambda_{2}:=\frac{\frac{2\underline{m}}{n^{2}(b-a)}}{\liminf_{\xi\rightarrow+\infty}\frac{\int^{b}_{a}\sup_{t\in Q(\xi)}F(x,t)dx}{\xi^{2}}}, \end{aligned} \] for every non-negative function \(G :[a,b]\times\mathbb R^{n}\rightarrow\mathbb R\), measurable in \([a,b]\), \(C^{1}\) in \(\mathbb R^{n}\) and satisfying the condition \[ G_{\infty}:=\lim_{\xi\rightarrow+\infty}\frac{\int^{b}_{a}\sup_{t\in Q(\xi)}F(x,t)dx}{\xi^{2}}<+\infty, \] and for \(\mu\in(0,\mu_{G,\lambda})\), where \[ \mu_{G,\lambda}:=\frac{2\underline{m}}{n^{2}(b-a)G_{\infty}}\left(1-\lambda\frac{n^{2}(b-a)}{2\underline{m}}\liminf_{\xi\rightarrow+\infty}\frac{\int^{b}_{a}\sup_{t\in Q(\xi)}F(x,t)dx}{\xi^{2}}\right), \] system (0.1) has an unbounded sequence of weak solutions in \((W^{1,2}_{0}([a,b]))^{n}\).
The author presents some examples to illustrate the results.