On the solvability of an indefinite nonlinear Kirchhoff equation via associated eigenvalue problems

We study the non-existence, existence and multiplicity of positive solutions to the following nonlinear Kirchhoff equation:% \begin{equation*} \left\{ \begin{array}{l} -M\left( \int_{\mathbb{R}^{3}}\left\vert \nabla u\right\vert ^{2}dx\right) \Delta u+\mu V\left( x\right) u=Q(x)\left\vert u\right\vert ^{p-2}u+\lambda f\left( x\right) u\text{ in }\mathbb{R}^{N}, \\ u\in H^{1}\left( \mathbb{R}^{N}\right) ,% \end{array}% \right. \end{equation*}% where $N\geq 3,2<p<2^{\ast }:=\frac{2N}{N-2},M\left( t\right) =at+b$ $\left( a,b>0\right) ,$ the potential $V$ is a nonnegative function in $\mathbb{R}$ and the weight function $Q\in L^{\infty }\left( \mathbb{R}^{N}\right)$ with changes sign in $\overline{\Omega }:=\left\{ V=0\right\} .$ We mainly prove the existence of at least two positive solutions in the cases that $$ $2<p<\min \left\{ 4,2^{\ast }\right\}$ and $0<\lambda <$ $\left( ii\right)$ $p\geq 4,\lambda \geq \lambda _{1}\left( f_{\Omega }\right)$ and near $\lambda _{1}\left( f_{\Omega }\right)$ for $\mu >0$ sufficiently large, where $\lambda _{1}\left( f_{\Omega }\right)$ is the first eigenvalue of $-\Delta$ in $$ with weight function f_{\Omega }:=f|_{% \overline{\Omega }}, whose corresponding positive principal eigenfunction is denoted by $\phi _{1}.$ Furthermore, we also investigated the non-existence and existence of positive solutions if $a,\lambda$ belongs to different intervals.