关键词: elliptic equations, bi-harmonic operator, normalized solutions, profile decomposition

Abstract: 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. Denoting 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 $\mu\in ( -\lambda_0, \infty)$ 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 a certain mass level below $\|Q^*\|_{L^2}$, provided that $\mu\in (-\lambda_1,0)$, where $\lambda_1=\frac{4\|\nabla Q^*\|^2_{L^2}}{\|Q^*\|^2_{L^2}}$ 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 us to fill the gap concerning the existence of the ground states for the BNLS when $\mu$ is negative and $p\in (1,1+\frac8d]$.

Key words: elliptic equations, bi-harmonic operator, normalized solutions, profile decomposition