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In this article we consider the Kirchhoff-type elliptic problem $$\displaylines{ -(a+b\int_{\Omega}|\nabla u|^2dx)\Delta u=|u|^{p-2}u, \quad\text{in } \Omega,\cr u=0, \quad \text{on } \partial\Omega, }$$ where $\Omega\subset\mathbb{R}^N$ and $p\in(2,2^*)$ with $2^*=\frac{2N}{N-2}$ if $N\geq 3$, and $2^*=+\infty$ otherwise. We show that the problem possesses infinitely many sign-changing solutions by using combination of invariant sets of descent flow and the Ljusternik-Schnirelman type minimax method.