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

Sheng-Sen Lu

Published 2015-10-25Version 1

We consider the following autonomous Kirchhoff-type equation $$-\left(a+b\int_{\mathbb{R}^N}|\nabla{u}|^2\right)\Delta{u}= f(u),~~~~u\in H^1(\mathbb{R}^N),$$ where $a\geq0,b>0$ are constants and $N\geq1$. Under general assumptions on the nonlinearity $f$, we establish the existence results of a ground state and multiple 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 and a new description of the critical values in association with the level argument.