We investigate the following fractional $p$-Laplacian equation \[ \begin{ cases} \begin{ aligned} (-\Delta)_p^s u&=\lambda |u|^{ q-2}u+|u|^{ p_s^*-2}u &&\text{ in}~\Omega,\\ u &=0 &&\text{ in}~ \mathbb{ R}^n\setminus\Omega, \end{ aligned} \end{ cases} \] where $s\in (0,1)$, $p>q>1$, $n>sp$, $\lambda>0$, $p_s^*=\frac{ np}{ n-sp}$ and $\Omega$ is a bounded domain (with $C^{ 1, 1}$ boundary). Firstly, we get a dichotomy result for the existence of positive solution with respect to $\lambda$. For $p\ge 2$, $p-1<q<p$, $n>\frac{ sp(q+1)}{ q+1-p}$, we provide two positive solutions for small $\lambda$. Finally, without sign constraint, for $\lambda$ sufficiently small, we show the existence of infinitely many solutions.
Categories: math.AP