An autonomous Kirchhoff-type equation with general nonlinearity in RN

We consider the following autonomous Kirchhoff-type equation −(a+b∫RN|∇u|2)Δu=f(u),u∈H1(RN), where a≥0,b>0 are constants and N≥1. 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≥2, 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.