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

Abstract

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 $p \in \left( {1,1 + {8 \over d}} \right)$, there exist orbit ally stable ground state solutions for the BNLS when μ ∈ (−λ₀, ∞) for some ${\lambda _0} = {\lambda _0}\left( {p,d,{{\left\| {{Q_p}} \right\|}_{{L^2}}}} \right) > \,0$. Moreover, in the mass-critical case $p = 1 + {8 \over d}$, we prove the orbital stability on a certain mass level below ${\left\| {{Q^ \ast }} \right\|_{{L^2}}}$ provided that μ ∈ (−λ₁, 0), where ${\lambda _1} = {{4\left\| {\nabla {Q^ \ast }} \right\|_{{L^2}}^2} \over {\left\| {{Q^ \ast }} \right\|_{{L^2}}^2}}$ 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 μ is negative and $p \in \left( {1,1 + {8 \over d}} \right]$.