  In this work we study the following nonlocal problem
  \begin{equation*}
  \left\{
  \begin{aligned}
  M(\|u\|^2_X)(-\Delta)^s u&= \lambda {f(x)}|u|^{\gamma-2}u+{g(x)}|u|^{p-2}u
&&\mbox{in}\ \ \Omega,
  u&=0 &&\mbox{on}\ \ \mathbb R^N\setminus \Omega,
  \end{aligned}
  \right.
  \end{equation*}
  where $\Omega\subset \mathbb R^N$ is open and bounded with smooth boundary,
$N>2s, s\in (0, 1), M(t)=a+bt^{\theta-1},\;t\geq0$ with $ \theta>1, a\geq 0$
and $b>0$. The exponents satisfy $1<\gamma<2<{2\theta<p<2^*_{s}=2N/(N-2s)}$
(when $a\neq 0$) and $2<\gamma<2\theta<p<2^*_{s}$ (when $a=0$). The parameter
$\lambda$ involved in the problem is real and positive. The problem under
consideration has nonlocal behaviour due to the presence of nonlocal fractional
Laplacian operator as well as the nonlocal Kirchhoff term $M(\|u\|^2_X)$, where
$\|u\|^{2}_{X}=\iint_{\mathbb R^{2N}}
\frac{|u(x)-u(y)|^2}{\left|x-y\right|^{N+2s}}dxdy$. The weight functions $f,
g:\Omega\to \mathbb R$ are continuous, $f$ is positive while $g$ is allowed to
change sign. In this paper an extremal value of the parameter, a threshold to
apply Nehari manifold method, is characterized variationally for both
degenerate and non-degenerate Kirchhoff cases to show an existence of at least
two positive solutions even when $\lambda$ crosses the extremal parameter value
by executing fine analysis based on fibering maps and Nehari manifold.
