On indefinite Kirchhoff-type equations under the combined effect of linear and superlinear terms

We investigate a class of Kirchhoff type equations involving a combination of linear and superlinear terms as follows: begin{equation*} -left( aint_{mathbb{R}^{N}}| abla u|^{2}dx+1right) Delta u+mu V(x)u=lambda f(x)u+g(x)|u|^{p-2}uquad text{ in }mathbb{R}^{N}, end{equation*}% where $Ngeq 3,2<p<2^{ast }:=frac{2N}{N-2}$, $Vin C(mathbb{R}^{N})$ is a potential well with the bottom $Omega :=int{xin mathbb{R}^{N} | V(x)=0}$. When $N=3$ and $4<p<6$, for each $a>0$ and $mu $ sufficiently large, we obtain that at least one positive solution exists for $% 0<lambdaleqlambda _{1}(f_{Omega}) $ while at least two positive solutions exist for $lambda _{1}(f_{Omega })< lambda<lambda _{1}(f_{Omega})+delta_{a}$ without any assumption on the integral $% int_{Omega }g(x)phi _{1}^{p}dx$, where $lambda _{1}(f_{Omega })>0$ is the principal eigenvalue of $-Delta $ in $H_{0}^{1}(Omega )$ with weight function $f_{Omega }:=f|_{Omega }$, and $phi _{1}>0$ is the corresponding principal eigenfunction. When $Ngeq 3$ and $2<p<min {4,2^{ast }}$, for $% mu $ sufficiently large, we conclude that $(i)$ at least two positive solutions exist for $a>0$ small and $0<lambda <lambda _{1}(f_{Omega })$; $% (ii)$ under the classical assumption $int_{Omega }g(x)phi _{1}^{p}dx<0$, at least three positive solutions exist for $a>0$ small and $lambda _{1}(f_{Omega })leq lambda<lambda _{1}(f_{Omega})+overline{delta }% _{a} $; $(iii)$ under the assumption $int_{Omega }g(x)phi _{1}^{p}dx>0$, at least two positive solutions exist for $a>a_{0}(p)$ and $lambda^{+}_{a}< lambda<lambda _{1}(f_{Omega})$ for some $a_{0}(p)>0$ and $lambda^{+}_{a}geq0$.