This paper is concerned with the existence and nonexistence of the minimizer of a $ L^2- $constraint minimization problem:$ \begin{align*} e(\alpha) = \inf\left\{E(u):u\in H, \int_{\mathbb{R}^N}u^2|\nabla u|^2\alpha_0 $ and $ e(\alpha) $ is not attained if $ 00 $ are also discussed.
