We study the non-existence, existence and multiplicity of positive solutions to the following nonlinear Kirchhoff equation:% begin{equation*} left{ begin{array}{l} -Mleft( int_{mathbb{R}^{3}}leftvert abla urightvert ^{2}dxright) Delta u+mu Vleft( xright) u=Q(x)leftvert urightvert ^{p-2}u+lambda fleft( xright) utext{ in }mathbb{R}^{N}, uin H^{1}left( mathbb{R}^{N}right) ,% end{array}% right. end{equation*}% where $Ngeq 3,2<p<2^{ast }:=frac{2N}{N-2},Mleft( tright) =at+b$ $left( a,b>0right) ,$ the potential $V$ is a nonnegative function in $mathbb{R}% ^{N}$ and the weight function $Qin L^{infty }left( mathbb{R}^{N}right) $ with changes sign in $overline{Omega }:=left{ V=0right} .$ We mainly prove the existence of at least two positive solutions in the cases that $% left( iright) $ $2<p<min left{ 4,2^{ast }right} $ and $0<lambda <% left[ 1-2left[ left( 4-pright) /4right] ^{2/p}right] lambda _{1}left( f_{Omega }right) ;$ $left( iiright) $ $pgeq 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.
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