  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 $\mathbb{R}%
^{N}$ 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 $%
\left( i\right) $ $2<p<\min \left\{ 4,2^{\ast }\right\} $ 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) $ 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 $% H_{0}^{1}\left( \Omega \right) $
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.
