We prove the existence of infinitely many solutions$u\in W_{0}^{1,2}(\unicode[STIX]{x1D6FA})$for the Kirchhoff equation$$\begin{eqnarray}\displaystyle -\biggl(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}\int _{\unicode[STIX]{x1D6FA}}|\unicode[STIX]{x1D6FB}u|^{2}\,dx\biggr)\unicode[STIX]{x1D6E5}u=a(x)|u|^{q-1}u+\unicode[STIX]{x1D707}f(x,u)\quad \text{in }\unicode[STIX]{x1D6FA}, & & \displaystyle \nonumber\end{eqnarray}$$where$\unicode[STIX]{x1D6FA}\subset \mathbb{R}^{N}$is a bounded smooth domain,$a(x)$is a (possibly) sign-changing potential,$0<q<1$,$\unicode[STIX]{x1D6FC}>0$,$\unicode[STIX]{x1D6FD}\geq 0$,$\unicode[STIX]{x1D707}>0$and the function$f$has arbitrary growth at infinity. In the proof, we apply variational methods together with a truncation argument.