### 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.

\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.

#### Keywords

Fractional Kirchhoff type problem; integrodifferential operator; truncation technique

#### Full Text:

PDFISSN: 1331-0623 (Print), 1848-8013 (Online)