Using variational methods and depending on a parameter$\unicode[STIX]{x1D706}$we prove the existence of solutions for the following class of nonlocal boundary value problems of Kirchhoff type defined on an exterior domain$\unicode[STIX]{x1D6FA}\subset \mathbb{R}^{3}$:$$\begin{eqnarray}\left\{\begin{array}{@{}ll@{}}M(\Vert u\Vert ^{2})[-\unicode[STIX]{x1D6E5}u+u]=\unicode[STIX]{x1D706}a(x)g(u)+\unicode[STIX]{x1D6FE}|u|^{4}u\quad & \text{in }\unicode[STIX]{x1D6FA},\\ u=0\quad & \text{on }\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA},\end{array}\right.\end{eqnarray}$$for the subcritical case ($\unicode[STIX]{x1D6FE}=0$) and also for the critical case ($\unicode[STIX]{x1D6FE}=1$).