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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mfenced open="{" close=""> <m:mrow> <m:mtable displaystyle="true"> <m:mtr> <m:mtd columnalign="left"> <m:mi>i</m:mi> <m:msub> <m:mrow> <m:mo>∂</m:mo> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mn>2</m:mn> <m:mi>V</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mn>2</m:mn> <m:mi mathvariant="normal">Ω</m:mi> <m:msub> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>z</m:mi> </m:mrow> </m:msub> <m:mi>u</m:mi> <m:mo>−</m:mo> <m:mi>λ</m:mi> <m:mi>u</m:mi> <m:mo>−</m:mo> <m:mi>b</m:mi> <m:mi>u</m:mi> <m:munder> <m:mrow> <m:mrow> <m:mstyle displaystyle="true"> <m:mo>∫</m:mo> </m:mstyle> </m:mrow> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:mrow> </m:munder> <m:mfrac> <m:mrow> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mi>u</m:mi> <m:mrow> <m:mrow> <m:mo>(</m:mo> </m:mrow> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mi>x</m:mi> <m:mo>−</m:mo> <m:mi>y</m:mi> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:mfrac> <m:mi mathvariant="normal">d</m:mi> <m:mi>y</m:mi> <m:mo>,</m:mo> <m:mspace width="1em"/> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">R</m:mi> <m:mo>×</m:mo> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:mspace width="1.0em"/> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:mi>u</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mspace width="1.0em"/> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> \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 <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xli