In this paper, we establish the multiplicity of nonnegative solutions to the following $p$-fractional Laplacian problem \[ \begin{cases} (-\Delta)^s_pu=f(x,u)+ambda g(x,u)ext{ in }mega,\quad u>0, u=0ext{ on }\mathbb R^n\backslashmega, \end{cases} \] where $\Omega$ is a smooth bounded set in $\mathbb R^n$, $n>ps$ with $s\in(0,1)$, $\lambda$ is a positive parameter, $f,g$ are homogeneous positive functions of degrees $q$ and $r$, respectively. Using fibering maps and the Nehari manifold, we obtain some results in subcritical and critical cases.