A Nehari manifold for non-local elliptic operator with concave-convex non-linearities and sign-changing weight function

In this article, we study the existence and multiplicity of non-negative solutions of following $p$-fractional equation: $$\{\begin{array}{l}\text{ds}-2{\int }_{\text{mb}{R}^n}\frac{\mathrm{\mid }u(y)-u(x){\mathrm{\mid }}^{p-2}(u(y)-u(x))}{\mathrm{\mid }x-y{\mathrm{\mid }}^{n+p\text{al}}}dxdy=\text{la}h(x)\mathrm{\mid }u{\mathrm{\mid }}^{q-1}u+b(x)\mathrm{\mid }u{\mathrm{\mid }}^{r-1}u\text{in}\text{Om}u\ge 0in\text{Om},u\in W^{\text{al},p}(\text{mb}{R}^n),u=0\text{on}\text{mb}{R}^n\setminus \text{Om}\end{array}$$ where $\text{Om}$ is a bounded domain in $\text{mb}{R}^n$, $p\geq 2$,$n> p\al$,$\al\in(0,1)$,$0< q<p-1 <r < \frac{np}{n-ps}-1$,$\la>0$and$h$,$b$ are sign changing smooth functions. We show the existence of solutions by minimization on the suitable subset of Nehari manifold using the fibering maps. We find that there exists ${\text{la}}_0$ such that for $\text{la}\in (0,{\text{la}}_0)$, it has at least two solutions.