On stability and instability of the ground states for the focusing inhomogeneous NLS with inverse-square potential
In this paper, we study the stability and instability of the ground states for the focusing inhomogeneous nonlinear Schrödinger equation with inverse-square potential (for short, INLS$_c$ equation): \[iu_{t} +Δu+c|x|^{-2}u+|x|^{-b} |u|^{σ} u=0,\; u(0)=u_{0}(x) \in H^{1},\;(t,x)\in \mathbb R\times\mathbb R^{d},\] where $d\ge3$, $0<b<2$, $0<σ<\frac{4-2b}{d-2}$ and $c\neq 0$ be such that $c<c(d):=\left(\frac{d-2}{2}\right)^{2}$. In the mass-subcritical case $0<σ<\frac{4-2b}{d}$, we prove the stability of the set of ground states for the INLS$_{c}$ equation. In the mass-critical case $σ=\frac{4-2b}{d}$, we first prove that the solution of the INLS$_c$ equation with initial data $u_{0}$ satisfying $E(u_0)<0$ blows up in finite or infinite time. Using this fact, we then prove that the ground state standing waves are unstable by blow-up. In the intercritical case $\frac{4-2b}{d}<σ<\frac{4-2b}{d-2}$, we finally show the instability of ground state standing waves for the INLS$_c$ equation.