On the inhomogeneous NLS with inverse-square potential

Abstract

We consider the inhomogeneous nonlinear Schrödinger equation with inverse-square potential in ${\mathbb{R}}^N$ $$i{u}_t+{\mathcal{L}}_{a}u+\lambda |x{|}^{-b}|u{|}^{\alpha }u=0,{\mathcal{L}}_a=\Delta -\frac{a}{|x{|}^2},$$ 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 $H_a^1({\mathbb{R}}^N)$ for $\lambda =1$, using a Gagliardo-Nirenberg-type estimate. In the sequel, we study local and global well-posedness in $H_a^1({\mathbb{R}}^N)$ 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 $H_a^1({\mathbb{R}}^N)$, for the mass-supercritical and energy-subcritical case.

arXiv comment27 pages

References in corpus (5)

• Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities
• Scattering of radial solutions to the Inhomogeneous Nonlinear Schrödinger Equation
• Rotating vortex clusters in media with inhomogeneous defocusing nonlinearity
• Focusing NLS with inverse square potential
• On well-posedness and concentration of blow-up solutions for the intercritical inhomogeneous NLS equation