A multiplicity result for a fractional Kirchhoff equation in $\mathbb{R}^{N}$ with a general nonlinearity

TL;DR

This paper proves the existence of multiple solutions for a fractional Kirchhoff equation in \\mathbb{R}^{N} with a general nonlinearity, using minimax methods under small parameter conditions.

Contribution

It introduces a multiplicity result for a fractional Kirchhoff equation with general nonlinearity, extending previous work to a broader class of problems.

Findings

Multiple solutions are established for the fractional Kirchhoff equation.

The results hold when the parameter q is sufficiently small.

The approach uses minimax arguments to prove existence.

Abstract

In this paper we deal with the following fractional Kirchhoff equation \begin{equation*} \left(p+q(1-s) \iint_{\mathbb{R}^{2N}} \frac{|u(x)- u(y)|^{2}}{|x-y|^{N+2s}} \, dx\,dy \right)(-\Delta)^{s}u = g(u) \mbox{ in } \mathbb{R}^{N}, \end{equation*} where $s\in (0,1)$, $N\geq 2$, $p>0$, $q$ is a small positive parameter and $g: \mathbb{R}\rightarrow \mathbb{R}$ is an odd function satisfying Berestycki-Lions type assumptions. By using minimax arguments, we establish a multiplicity result for the above equation, provided that $q$ is sufficiently small.