Multiplicity Results of a Nonlocal Problem Involving Concave-Convex Nonlinearities

In this work, we investigate the following fractional $p$-Laplacian equation involving a concave-convex nonlinearities as follows,
$$ {(P_\lambda)} \begin{cases} (-\Delta)_p^s u = \lambda u^{q} + u^{r} &\text{in }\Omega, \\ u>0 & \text{in }\Omega, \\ u = 0 &\text{in }\mathbb{R}^N\setminus\Omega, \end{cases} $$
where $\Omega\subset\mathbb{R}^N$, $N\geq 2$ is a bounded domain with $C^{1,1}$ boundary $\partial\Omega$, $\lambda >0$, $1<p<\infty$, $s\in (0,1)$ such that $N\geq s p$, $0<q<p-1<r\leq p^*_s-1$, $p^*_s = \frac{Np}{N-s p}$ is the fractional critical Sobolev exponent and the nonlinear nonlocal operator $(-\Delta)^s_p u$ with $s\in (0,1)$ is the $p$-fractional Laplacian defined on smooth functions by
\begin{align*} (-\Delta)^s_p u(x)=2 \underset{\epsilon\searrow 0}{\lim}\int_{\mathbb{R}^{N}\setminus B_\epsilon} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ ps}}\,{\mathrm d}y,\qquad x\in \mathbb{R}^N. \end{align*}
We use variational methods, in order to show the existence of multiple positive solutions to the problem $(P_\lambda)$ for different value of $\lambda$.