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

Abstract

We investigate the following fractional $p$-Laplacian equation \[ \begin{cases} \begin{aligned} (-Δ)_p^s u&=λ|u|^{q-2}u+|u|^{p_s^*-2}u &&\text{in}~Ω,\\ u &=0 &&\text{in}~ \mathbb{R}^n\setminusΩ, \end{aligned} \end{cases} \] where $s\in (0,1)$, $p>q>1$, $n>sp$, $\lambda >0$, $p_s^{\ast }=\frac{np}{n-sp}$ 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.