Orbital Stability of Standing Waves for a fourth-order nonlinear Schrödinger equation with the mixed dispersions

Abstract

In this paper, we study the ground state standing wave solutions for the focusing bi-harmonic nonlinear Schrödinger 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 $μ\in ( -λ_0, \iy)$ for some ${\lambda }_0={\lambda }_0(p,d,\Vert Q_p{\Vert }_{L^2})>0$. Moreover, in the mass-critical case $p=1+\frac{8}{d}$\,, we prove the orbital stability on certain mass level below $\Vert Q^{\ast }{\Vert }_{L^2}$, provided $μ\in (-\lam_1,0)$, where $\lam_1=\dfrac{4\|\nabla Q^*\|^2_{L^2}}{\|Q^*\|^2_{L^2}}$ and $Q^{\ast }=Q_{1+8/d}$. 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 $p\in (1,1+\frac{8}{d}]$.