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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 boundeddomain (with $C^{1, 1}$ boundary). Firstly, we get a dichotomy result for theexistence 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 forsmall $\lambda$. Finally, without sign constraint, for $\lambda$ sufficientlysmall, we show the existence of infinitely many solutions.

Existence and Multiplicity of Solutions for Fractional $p$-Laplacian Equation Involving Critical Concave-convex Nonlinearities