Multiple solutions for a Kirchhoff-type equation with general nonlinearity

Sheng-Sen Lu

Published 2016-02-03Version 1

This paper is devoted to the study of the following autonomous Kirchhoff-type equation $$-M\left(\int_{\mathbb{R}^N}|\nabla{u}|^2\right)\Delta{u}= f(u),~~~~u\in H^1(\mathbb{R}^N),$$ where $M$ is a continuous non-degenerate function and $N\geq2$. Under suitable additional conditions on $M$ and very general assumptions on the nonlinearity $f$, we establish certain existence results of multiple solutions by variational methods, which are also naturally interpreted from a non-variational point of view.