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 $N\geq 3,2<p<2^{\ast }:=\frac{2N}{N-2}$, $V\in C(\mathbb{R}^{N})$ is a potential well with the bottom $\Omega :=int\{x\in \mathbb{R}^{N}\ |\ V(x)=0\}$. 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 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 $$, where $\lambda _{1}(f_{\Omega })>0$ is the principal eigenvalue of $-\Delta$ in $H_{0}^{1}(\Omega )$ with weight function $f_{\Omega }:=f|_{\Omega }$, and $\phi _{1}>0$ is the corresponding principal eigenfunction. When $N\geq 3$ and $2<p<\min \{4,2^{\ast }\}$, for $$ sufficiently large, we conclude that $(i)$ at least two positive solutions exist for $a>0$ small and $0<\lambda <\lambda _{1}(f_{\Omega })$; $$ under the classical assumption $\int_{\Omega }g(x)\phi _{1}^{p}dx<0$, at least three positive solutions exist for $a>0$ small and $\lambda _{1}(f_{\Omega })\leq \lambda<\lambda _{1}(f_{\Omega})+\overline{\delta }$; $(iii)$ 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 $\lambda^{+}_{a}\geq0$.