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

TL;DR

This paper proves the existence of at least one non-trivial solution for a Kirchhoff type elliptic equation in R^N, employing a simple proof and variational methods under general nonlinear assumptions.

Contribution

It provides a straightforward proof of solution existence for Kirchhoff equations with nonlinearities satisfying Berestycki-Lions conditions, including the existence of ground states.

Findings

Existence of at least one non-trivial solution established.

Positive solutions are obtained under general nonlinear assumptions.

Ground states are identified using variational minimization.

Abstract

In this paper we present a very simple proof of the existence of at least one non trivial solution for a Kirchhoff type equation on $\RN$, for $N\ge 3$. 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.