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

Abstract

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}^N\backslash\Omega,
\end{equation*}
where $N>2s$ with $s\in(0,1)$, $\Omega$ 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.