This paper is devoted to considering the attainability of minimizers of the $L^2$-constraint variational problem

$$m_{\gamma ,a}=inf\{J_{\gamma }(u):u\in H^2({\mathbb{R}}^N),{\int }_{{\mathbb{R}}^N}|u{|}^{2}dx=a^2\},$$

where

$$J_{\gamma }(u)=\frac{\gamma }{2}{\int }_{{\mathbb{R}}^N}|\mathrm{\Delta }u{|}^{2}dx+\frac{1}{2}{\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }u{|}^{2}dx+\frac{1}{2}{\int }_{{\mathbb{R}}^N}V(x)|u{|}^{2}dx-\frac{1}{2\sigma +2}{\int }_{{\mathbb{R}}^N}|u{|}^{2\sigma +2}dx,$$

$\gamma >0$, $a>0$, $\sigma \in (0,\frac{2}{N})$ with $N\ge 2$. Moreover, the function $V:{\mathbb{R}}^N\to [0,+\mathrm{\infty })$ is continuous and bounded. By using the variational methods, we can prove that, when $V$ satisfies four different assumptions, $m_{\gamma ,a}$ are all achieved.