  We consider the following autonomous Kirchhoff-type equation
\begin{equation*} -\left(a+b\int_{\mathbb{R}^N}|\nabla{u}|^2\right)\Delta u=
f(u),~~~~u\in H^1(\mathbb{R}^N), \end{equation*} where $a\geq0,b>0$ are
constants and $N\geq1$. Under general Berestycki-Lions type assumptions on the
nonlinearity $f$, we establish the existence results of a ground state and
multiple radial solutions for $N\geq2$, and obtain a nontrivial solution and
its uniqueness, up to a translation and up to a sign, for $N=1$. The proofs are
mainly based on a rescaling argument, which is specific for the autonomous
case, and a new description of the critical values in association with the
level sets argument.
