The elliptic Kirchhoff equation in $\mathbb R^N$ perturbed by a local nonlinearity

In this paper we present a very simple proof of the existence of at least one nontrivial solution for a Kirchhoff-type equation on R^N, for N \geq 3. In particular, in the first part of the paper we are interested in studying the existence of a positive solution to the elliptic Kirchhoff equation under the effect of a nonlinearity satisfying the general Berestycki-Lions assumptions. In the second part we look for ground states using minimizing arguments on a suitable natural constraint.