An autonomous Kirchhoff-type equation with general nonlinearity in $\mathbb{R}^N$

TL;DR

This paper investigates the existence and multiplicity of solutions for an autonomous Kirchhoff-type equation with general nonlinearity in , using rescaling and level set techniques, providing new insights into solution structures.

Contribution

It introduces novel methods for analyzing solution existence and multiplicity for Kirchhoff equations with general nonlinearities, including a new description of critical values.

Findings

Existence of a ground state for N 

Multiple radial solutions for N 

Uniqueness of solutions for N=1

Abstract

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.