Normalized solutions for the mixed dispersion nonlinear Schrödinger equations with four types of potentials and mass subcritical growth

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}} \vert u\vert^2 dx = a^2 \} {, } $ where $ J_{\gamma}(u) = \frac{\gamma}{2}\int_{\mathbb{R}^{N}} \vert\Delta u\vert^2 dx+\frac{1}{2}\int_{\mathbb{R}^{N}} \vert\nabla u\vert^2 dx+\frac{1}{2}\int_{\mathbb{R}^{N}} V(x)\vert u\vert^2 dx-\frac{1}{2\sigma+2}\int_{\mathbb{R}^{N}} \vert u\vert^{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}\rightarrow [0, +\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.