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( a\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx+1\right) \Delta u+\mu V(x)u=\lambda f(x)u+g(x)|u|^{p-2}u\quad \text{ in }\mathbb{R}^{N}, \end{equation*}% where $N\geq 3,2 0$ and $\mu $ sufficiently large, we obtain that at least one positive solution exists for $% 0 0$ 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 $N\geq 3$ and $2 0$ small and $0 0$ small and $\lambda _{1}(f_{\Omega })\leq \lambda 0$, at least two positive solutions exist for $a>a_{0}(p)$ and $\lambda^{+}_{a} 0$ and $\lambda^{+}_{a}\geq0$.