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

We investigate the following fractional $p$-Laplacian equation $$\{\begin{array}{l}\begin{array}{rlrl}{\displaystyle (-\mathrm{\Delta }{)}_p^{s}u}& {\displaystyle =\lambda \mathrm{\mid }u{\mathrm{\mid }}^{q-2}u+\mathrm{\mid }u{\mathrm{\mid }}^{p_s^{\ast }-2}u}& {\displaystyle }& {\displaystyle \text{in }\mathrm{\Omega },}\\ {\displaystyle u}& {\displaystyle =0}& {\displaystyle }& {\displaystyle \text{in }{\mathbb{R}}^n\setminus \mathrm{\Omega },}\end{array}\end{array}$$ where $s\in (0,1)$, $p>q>1$, $n>sp$, $\lambda >0$, $p_s^{\ast }=\frac{np}{n-sp}$ and $\mathrm{\Omega }$ 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.