Blow-up solutions to the Hartree-Fock type Schrödinger equation with the critical rotational speed

Abstract

In this article, we are concerned with the existence, non-existence, and blow-up behavior of normalized ground state solutions for the mass critical Hartree-Fock type Schrödinger equation with rotation i∂tu=−Δu+2V(x)u+2ΩLzu−λu−bu∫RN∣u(y)∣2∣x−y∣2dy,(t,x)∈R×RN,u(0,x)=u0(x),\left\{\begin{array}{l}i{\partial }_{t}u=-\Delta u+2V\left(x)u+2\Omega {L}_{z}u-\lambda u-bu\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{{| u(y)| }^{2}}{{| x-y| }^{2}}{\rm{d}}y,\hspace{1em}\left(t,x)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N},\hspace{1.0em}\\ u\left(0,x)={u}_{0}\left(x),\hspace{1.0em}\end{array}\right. where N≥3N\ge 3, b>0b\gt 0, V(x)=∣x∣22V\left(x)=\frac{{| x| }^{2}}{2}, Lz{L}_{z} is the angular momentum operator with the critical rotational speed Ω=1\Omega =1, and the constant λ\lambda is the unknown Lagrange multiplier. We prove that the L2{L}^{2}-constraint minimizers exist if and only if the parameter bb satisfies b<b*=‖U‖22b\lt {b}_{* }={\Vert U\Vert }_{2}^{2}, where UU is a positive radially symmetric ground state of −Δu+u−u∫RNu2(y)∣x−y∣2dy=0-\Delta u+u-u{\int }_{{{\mathbb{R}}}^{N}}\frac{{u}^{2}(y)}{{| x-y| }^{2}}{\rm{d}}y=0 in RN{{\mathbb{R}}}^{N}. We also establish the orbital stability result of prescribed mass standing waves for the equation when b<b*b\lt {b}_{* }. When bb approaches b*{b}_{* }, the system collapses to a profile obtained from the optimizer of a Gagliardo-Nirenberg inequality.

Keywords

• hartree-fock type schrödinger equation
• normalized solutions
• rotation
• blow-up solutions
• 35j61
• 35j20
• 35q55
• 49j40