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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 underconsideration has nonlocal behaviour due to the presence of nonlocal fractionalLaplacian 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 tochange sign. In this paper an extremal value of the parameter, a threshold toapply Nehari manifold method, is characterized variationally for bothdegenerate and non-degenerate Kirchhoff cases to show an existence of at leasttwo positive solutions even when $\lambda$ crosses the extremal parameter valueby executing fine analysis based on fibering maps and Nehari manifold.

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