  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 $\mu\in ( -\lambda_0, \iy)$ for some $\lambda_0=\lambda_0(p,
d,\|Q_p\|_{L^2})>0$. Moreover, in the mass-critical case $p=1+\frac{8}{d}$\,,
we prove the orbital stability on certain mass level below $\|Q^*\|_{L^2}$,
provided $\mu\in (-\lam_1,0)$, 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 $p\in (1,1+\frac8d]$.

Analysis of PDEs