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In this work, we consider the inhomogeneous nonlinear Schr\"odinger (INLS)equation in $\mathbb{R}^n$ \begin{align} i\partial_t u + \Delta u + \gamma|x|^{-b}|u|^{\alpha} u = 0, \end{align} where $\gamma=\pm 1$, and $\alpha$ and$b$ are positive numbers. Our main focus is to estabilish the globalwell-posedness of the INLS equation in Lorentz spaces for $0<b<2$ and$\alpha<\frac{4-2b}{N-2}$. To achieve this, we use Strichartz estimates inLorentz spaces $L^{r,q}(\R^n)$ combined with a fixed point argument. Working onLorentz space setting instead the classical $L^p$ is motivated by the fact thatthe potential $|x|^{-b}$ does not belong the usual $L^p$-space. As aconsequence of the ideas developed here on the global solution study we obtainsome other properties for INLS, such as, existence of self-similar solutions,scattering, wave operators and assymptotic stability.

On global well-posedness, scattering and other properties for infinity energy solutions to inhomogeneous NLS Equation