Inhomogeneous Schrödinger system with inverse square potential in three space dimensions

This work studies the inhomogeneous Schrödinger coupled system with inverse square potential $$ i\partial _{t} u_{j} +\Delta u_{j}-\frac{\lambda}{|x|^{2}}u_{j} =\pm |x|^{- \tau}\Big(\displaystyle \sum _{k=1}^{m}a_{jk}|u_{k}|^{p}\Big)|u_{j}|^{p-2}u_{j}. $$ In this paper, $u_{j}:\mathbb{R}\times \mathbb{R}^{3}\to \mathbb{C}$ and the above parameters satisfy $1\leq j\leq m$ , $\tau >0$ , $1\leq p<3-\tau $ and $\lambda >-\frac{1}{4}$ . In order to avoid the singular term $|u_{j}|^{p-2}$ , one assumes that $p\geq 2$ . The invariant Sobolev norm under the classical scaling $\|\kappa ^{\frac{2-\tau}{2(p-1)}}{u_{j}}(\kappa ^{2} t,\kappa \cdot ) \|_{\dot{H}^{s_{c}}}=\|{u_{j}}(\kappa ^{2} t)\|_{\dot{H}^{s_{c}}}$ gives the energy critical exponent $3-\tau $ , which corresponds to $1=s_{c}:=\frac{3}{2}-\frac{2-\tau}{2(p-1)}$ . So, necessarily, one needs to assume that $0<\tau <1$ . The assumption $\lambda >-\frac{1}{4}$ is motivated by the critical Hardy inequality $\frac{1}{4}\int _{\mathbb{R}^{3}}\frac{|f(x)|^{2}}{|x|^{2}}\,dx\leq \int _{ \mathbb{R}^{3}}|\nabla f(x)|^{2}\,dx$ . In the stationary regime, one proves a Gagliardo-Nirenberg estimate adapted to the above problem and one establishes the existence of ground states. In the evolution regime, one develops an energy local theory. Moreover, for small datum, a local solution extends to a global one which scatter in the energy space. Eventually, in the inter-critical regime, one obtains a dichotomy of global existence versus blow-up of energy solutions under the ground state threshold. The limit case $\tau \to 0$ gives some results about the local/global well-posedness of the homogeneous Schrödinger system with inverse square potential in the energy space. Indeed, to the authors knowledge, such a problem was not treated before.