The Existence and Stability of Normalized Solutions for a Bi-Harmonic Nonlinear Schrödinger Equation with Mixed Dispersion

In this paper, we study the ground state standing wave solutions for the focusing bi-harmonic nonlinear Schrödinger equation with a μ-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 μ = 0, we prove that in the mass-subcritical regime p∈(1,1+8d), there exist orbit ally stable ground state solutions for the BNLS when μ ∈ (−λ₀, ∞) for some λ0=λ0(p,d,‖Qp‖L2)>0. Moreover, in the mass-critical case p=1+8d, we prove the orbital stability on a certain mass level below ‖Q∗‖L2 provided that μ ∈ (−λ₁, 0), where λ1=4‖∇Q∗‖L22‖Q∗‖L22 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 μ is negative and p∈(1,1+8d].