In this paper, we study the existence of nontrivial solution and nonnegative least energy solution for the following nonlinear Kirchhoff type elliptic equation
$\left\{ \begin{align} & -(a+b\int_{{{\mathbb{R}}^{3}}}{|}\nabla u{{|}^{2}}\text{d}x)\Delta u+V(x)u=\mu u+|u{{|}^{p-1}}u,\ \ \ \ \ \ \ \ \ \ x\in {{\mathbb{R}}^{3}}, \\ & u\in {{H}^{1}}({{\mathbb{R}}^{3}}),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in {{\mathbb{R}}^{3}}, \\ \end{align} \right.$
where $p\in (3, 5)$, $a, b>0$, $V\in C(\mathbb{R} ^3, \mathbb{R} ^+)$ and $\lim\limits_{|x|\to +\infty}V(x)=\infty$. By using variational methods, firstly we prove that for any $b>0$, there exists $\delta(b)>0$ such that problem (0.1) (0.1) with $\mu_1\leq\mu <\mu_1+\delta(b)$ has a nontrivial solution, where $\mu_1$ denotes the first eigenvalue of the Schrödinger operator $-\triangle+V$. Secondly, we show that there exists $\delta_1(b)\in(0, \delta(b))$ such that problem (0.1) (0.1) with $\mu_1 <\mu <\mu_1+\delta_1(b)$ has a nonnegative least energy solution. Finally, by using the symmetric Mountain Pass lemma we prove that problem (0.1) (0.1) has infinitely many nontrivial solutions for any $\mu\in \mathbb{R} $.
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