$\mathrm{L}^{2}_{\alpha} $-Solutions for Nonlinear Schr\"odinger Equations Associated With The Weinstein Operator

Abstract

In this paper, we study the Schr\"odinger equation associated with the Weinstein operators and we prove the existence and uniqueness of global solutions to Schr\"odinger-Weinstein equations in\\ $C\left(\left(-T_{\min }, T_{\max }\right) ; L_{\alpha}^{2}\left(\mathbb{R}^{d+1}_+\right)\right) \bigcap L_{l o c}^{q_{1}}\left(\left(-T_{\min }, T_{\max }\right) ; L_{\alpha}^{r_{1}}\left(\mathbb{R}^{d+1}_+\right)\right)$ \\ on initial condition in $L_{\alpha}^{2}\left(\mathbb{R}^{d+1}_+\right)$