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| In this paper, we study the existence of multiple positive solution for the following equation: \begin{equation} \label{eq0} \left\{ \begin{aligned} &-\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx\right)\Delta u+V_{\lambda}(x)u=f(x)|u|^{q-2}u+g(x)|u|^{p-2}u ,&x\in \mathbb{R}^{3},\\ &u>0,u\in H^{1}(\mathbb{R}^{3}), \end{aligned} \right. \end{equation} where $a,b>0$ are constants, $1<q<2$, $4<p<6$ and $V_{\lambda}(x)=\lambda V^{+} (x)-V^{-}(x)$, $V^{\pm}=\max\{\pm V,0\}\ne 0$, $\lambda>0$. Assume that the functions $V, f, g$ satisfy the following conditions: $(V_{1})$\ $V^{+}\in C(\mathbb{R}^{3})$, $V^{-}\in L^{\frac{3}{2}}(\mathbb{R}^{3})$ with $\|V^{-} \|_{L^{\frac{3}{2}}}<aS$, where $S$ is the best Sobolev imbedding constant of $D^{1,2}(\mathbb{R}^{3})$ into $L^{6}(\mathbb{R}^{3})$; $(V_{2})$\ there exists $k>0$ such that $\{V^{+}<k\}=\{x\in \mathbb{R}^{3}: V^{+}(x)<k\}$ is nonempty and has finite measure; $(V_{3})$\ $\Omega:=$int$((V^{+} )^{-1}(0)$ is nonempty and has a smooth boundary with $\bar{\Omega} :=(V^{+})^{-1}(0)$, where $(V^{+})^{-1}(0):=\{x\in \mathbb{R}^{3}: V^{+}(x)=0\}$; $(f)$\ $f\in L^{\frac{6}{6-q}}(\mathbb{R}^{3})$ with the set $\{x\in \mathbb{R}^{3}: f(x)>0\}$ of positive measure; $(g_{1})$\ $g\in L^{\frac{6}{6-p}}(\mathbb{R}^3)$ with the set $\{x\in\mathbb{R}^{3}: g(x)>0\}$ of positive measure; $(g_{2})$ there exists a nonempty open set $\Omega_{g}\subset \Omega$ such that $g>0$ a.e. on $\Omega_{g}$. $(fg)$\ there is $\mu _{0}>1$ such that $$ \|f^{+}\|_{L^{\frac{6}{6-q}}}^{p-2}\|g^{+}\|_{L^{\frac{6}{6-p}}}^{2-q}< \left(\frac{\mu _{0}-1}{2d_6^{2}\mu_{0}(p-q)}\right)^{p-q}(2-q)^{2-q}(p-2)^{p-2}, $$ where $d_6$ is a embedding constant given in $\eqref{eq5}$ and $\|\cdot\|_{L^{s}}$ denotes the norm of Lebesgue space $L^{s}({\mathbb{R}^{3}})$. |
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| Keywords:Basic mathematics；Kirchhoff type equation; Multiple solutions; Variational methods; Concave-convex nonlinearities |
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