Nehari manifold approach for fractional Kirchhoff problems with extremal value of the parameter

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.