Existence and multiplicity of solutions for a class of fractional Kirchhoff-type problem

In this paper, we establish the existence and multiplicity of solutions to the following fractional Kirchhoff-type problem begin{equation*}M(|u|^2)(-Delta)^s u=f(x,u(x)), mbox{ in } Omega ,,, u=0 mbox{ in } mathbb{R}^NbackslashOmega, end{equation*} where $N>2s$ with $sin(0,1)$, $Omega$ is an open bounded subset of $mathbb{R}^N$ with Lipschitz boundary, $M$ and $f$ are two continuous functions, and $(-Delta)^s$ is a fractional Laplace operator. Our main tools are based on critical point theorems and the truncation technique