On stability and instability of the ground states for the focusing inhomogeneous NLS with inverse-square potential

Abstract

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 $d\ge 3$, $0<b<2$, $0<\sigma <\frac{4-2b}{d-2}$ and $c\ne 0$ be such that $c<c(d):={\left(\frac{d-2}{2}\right)}^2$. In the mass-subcritical case $0<\sigma <\frac{4-2b}{d}$, we prove the stability of the set of ground states for the INLS${}_c$ equation. In the mass-critical case $\sigma =\frac{4-2b}{d}$, 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 $\frac{4-2b}{d}<\sigma <\frac{4-2b}{d-2}$, we finally show the instability of ground state standing waves for the INLS${}_c$ equation.

15 Pages