Multiple solutions for a Kirchhoff-type equation with general nonlinearity

TL;DR

This paper investigates the existence of multiple solutions to a Kirchhoff-type equation with general nonlinearities using variational methods, extending known results under broad conditions on the nonlinearity and the function M.

Contribution

It introduces new existence results for multiple solutions to Kirchhoff equations with general nonlinearities, under broad conditions on M and f, using variational techniques.

Findings

Established multiple solutions under Berestycki-Lions type conditions.

Extended solution existence results to non-variational frameworks.

Applied variational methods to a class of nonlinear Kirchhoff equations.

Abstract

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 general Berestycki-Lions type assumptions on the nonlinearity $f$, we establish several existence results of multiple solutions by variational methods, which are also naturally interpreted from a non-variational point of view.