Nehari manifold for fractional p(.)-Laplacian system involving concave-convex nonlinearities

In this article using Nehari manifold method we study the multiplicity of solutions of the following nonlocal elliptic system involving variable exponents and concave-convex nonlinearities: \begin{equation*} ;;; \begin{array}{rl} (-\Delta){p(\cdot)}^{s} u&=\lambda~ a(x)| u|^{q(x)-2}u+\frac{\alpha(x)}{\alpha(x)+\beta(x)}c(x)| u|^{\alpha(x)-2}u| v| ^{\beta(x)},\hspace{2mm} x\in \Omega; \ (-\Delta){p(\cdot)}^{s} v&=\mu~ b(x)| v|^{q(x)-2}v+\frac{\alpha(x)}{\alpha(x)+\beta(x)}c(x)| v|^{\alpha(x)-2}v| u| ^{\beta(x)},\hspace{2.5mm} x\in \Omega; \ u=v&=0 ,\hspace{1cm} x\in \Omega^c:=\mathbb R^N\setminus\Omega, \end{array} \end{equation*} where $\mathrm{\Omega }\subset {\mathbb{R}}^N,N\ge 2$ is a smooth bounded domain, $\lambda ,\mu >0$ are the parameters, $s\in (0,1),$ $p\in C(\mathbb R^N\times \mathbb R^N,(1,\infty))$and$q,\alpha,\beta\in C(\overline{\Omega},(1,\infty))$ are the variable exponents and $a,b,c\in C(\bar{\mathrm{\Omega }},[0,\mathrm{\infty }))$ are the non-negative weight functions. We show that there exists $\mathrm{\Lambda }>0$ such that for all $\lambda +\mu <\mathrm{\Lambda }$, there exist two non-trivial and non-negative solutions of the above problem under some assumptions on $q,\alpha ,\beta$.