We consider the inhomogeneous nonlinear Schrödinger equation with inverse-square potential in \({\mathbb {R}}^N\)

where \(\lambda =\pm 1\), \(\alpha ,b>0\) and \(a>-\frac{(N-2)^2}{4}\). We first establish sufficient conditions for global existence and blow-up in \(H^1_a({\mathbb {R}}^N)\) for \(\lambda =1\), using a Gagliardo–Nirenberg-type estimate. In the sequel, we study local and global well-posedness in \(H^1_a({\mathbb {R}}^N)\) in the \(H^1\)-subcritical case, applying the standard Strichartz estimates combined with the fixed point argument. The key to do that is to establish good estimates on the nonlinearity. Making use of these estimates, we also show a scattering criterion and construct a wave operator in \(H^1_a({\mathbb {R}}^N)\), for the mass-supercritical and energy-subcritical case.

我们考虑在\({\mathbb {R}}^N\) 中具有平方反比势的非齐次非线性薛定谔方程

其中\(\lambda =\pm 1\)、\(\alpha ,b>0\)和\(a>-\frac{(N-2)^2}{4}\)。我们首先使用 Gagliardo-Nirenberg 类型的估计为\(\lambda =1\)在\(H^1_a({\mathbb {R}}^N)\) 中建立全局存在和爆炸的充分条件。在续集中，我们研究了\(H^1_a({\mathbb {R}}^N)\)在\(H^1\) -亚临界情况下的局部和全局适定性，应用标准 Strichartz 估计组合与不动点参数。做到这一点的关键是建立对非线性的良好估计。利用这些估计，我们还展示了一个散射准则并在\(H^1_a({\mathbb {R}}^N)\)，对于质量-超临界和能量-亚临界情况。