Sharp threshold for global existence and blowup in the focusing nonlinear Schrödinger equation with inverse-square potential

This paper examines the focusing nonlinear Schrödinger equation with an inverse-square potential in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <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:mspace width="0.3333em"/> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>N</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> RN (N≥ 3) , where the nonlinear exponent lies between the mass-critical and energy-subcritical regimes. A central approach of this work is the construction of novel cross-invariant manifolds to address the challenge of proving blow-up within the potential well theory. Utilizing these manifolds, we establish a sharp threshold for the global existence and finite-time blow-up of weak solutions.