  By introducing some new tricks, we prove that the nonlinear problem of
Kirchhoff-type \begin{equation*} \left\{
  \begin{array}{ll}
  -\left(a+b\int_{\R^3}|\nabla u|^2\mathrm{d}x\right)\triangle u+V(x)u=f(u), &
x\in \R^3;
  u\in H^1(\R^3),
  \end{array} \right. \end{equation*} admits two class of ground state
solutions under the general "Berestycki-Lions assumptions" on the nonlinearity
$f$ which are almost necessary conditions, as well as some weak assumptions on
the potential $V$. Moreover, we also give a simple minimax characterization of
the ground state energy. Our results improve and complement previous ones in
the literature.
