Orbital Stability of Standing Waves for a fourth-order nonlinear Schr\"odinger equation with the mixed dispersions

In this paper, we study the ground state standing wave solutions for the focusing bi-harmonic nonlinear Schr"{o}dinger equation with a $\mu$-Laplacian term (BNLS). Such BNLS models the propagation of intense laser beams in a bulk medium with a second-order dispersion term. Denote by $Q_p$ the ground state for the BNLS with $\mu =0$. We prove that in the mass-subcritical regime $p\in (1,1+\frac{8}{d})$, there exist orbitally stable {ground state solutions} for the BNLS when $\mu \in (-{\lambda }_0,\text{iy})$ for some $\lambda_0=\lambda_0(p, d,|Q_p|{L^2})>0$\mathrm{.}Moreover,inthemass-criticalcase$p=1+\frac{8}{d}$,, we prove the orbital stability on certain mass level below $\mathrm{\parallel }{Q}^{\ast }{\mathrm{\parallel }}_{L^2}$, provided $\mu \in (-{\text{lam}}_1,0)$, where $\lam_1=\dfrac{4|\nabla Q^*|^2{L^2}}{|Q^|^2_{L^2}}$and$Q^=Q_{1+8/d}$. The proofs are mainly based on the profile decomposition and a sharp Gagliardo-Nirenberg type inequality. Our treatment allows to fill the gap concerning existence of the ground states for the BNLS when $\mu$ is negative and $p\in (1,1+\frac{8}{d}]$.