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 }$$\mathbb{R}$^{N}, \ u\in H^{1}\left( $\mathbb{R}$^{N}\right) ,% \end{array}% \right. \end{equation*}% where $N\ge 3,2<p<2^{\ast }:=\frac{2N}{N-2},M\left(t\right)=at+b$ $\left( a,b>0\right) ,$thepotential$V$isanonnegativefunctionin$$\mathbb{R}$% ^{N}$andtheweightfunction$Q\in L^{\infty }\left( $\mathbb{R}$^{N}\right) $withchangessignin$\overline{\Omega }:=\left{ V=0\right} .$ We mainly prove the existence of at least two positive solutions in the cases that $% \left( i\right) $ $2<p<\min \{4,2^{\ast }\}$ and $0<\lambda <% \left[ 1-2\left[ \left( 4-p\right) /4\right] ^{2/p}\right] \lambda {1}\left( f{\Omega }\right) ;$ $\left(ii\right)$ $p\geq 4,\lambda \geq \lambda {1}\left( f{\Omega }\right) $andnear$\lambda {1}\left( f{\Omega }\right) $for$\mu >0$sufficientlylarge,where$\lambda {1}\left( f{\Omega }\right) $isthefirsteigenvalueof$-\Delta $in$% H_{0}^{1}\left( \Omega \right) $withweightfunction$f_{\Omega }:=f|_{% \overline{\Omega }},$ whose corresponding positive principal eigenfunction is denoted by ${\varphi }_1\mathrm{.}$ Furthermore, we also investigated the non-existence and existence of positive solutions if $a,\lambda$ belongs to different intervals.