Existence and multiplicity results for fractional $p$-Kirchhoff equation with sign changing nonlinearities

Abstract

In this paper, we show the existence and multiplicity of nontrivial, non-negative solutions of the fractional $p$-Kirchhoff problem \begin{equation*} \begin{array}{rllll} M\left(\displaystyle\int_{\mathbb{R}^{2n}}\frac{|u(x)-u(y)|^p}{\left|x-y\right|^{n+ps}}dx\,dy\right)(-\Delta)^{s}_p u &=\lambda f(x)|u|^{q-2}u+ g(x)\left|u\right|^{r-2}u\, \text{in} \Omega,\\ u&=0 \;\mbox{in} \mathbb{R}^{n}\setminus \Omega, \end{array} \end{equation*} where $(-\Delta)^{s}_p$ is the fractional $p$-Laplace operator, $\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary, $f \in L^{\frac{r}{r-q}}(\Omega)$ and $g\in L^\infty(\Omega)$ are sign changing, $M$ is continuous function, $ps<n<2ps$ and $1<q<p<r\leq p_s^*=\frac{np}{n-ps}$.