Local well-posedness of the inhomogeneous nonlinear Schrödinger equation with Coulomb potential

In this work, we investigate an inhomogeneous nonlinear Schrödinger equation (INLS) that includes a Coulomb-type potential. Our primary objective is to establish a local well-posedness theory within the energy space, specifically focusing on the repulsive regime of the potential. In addition, we aim to develop local well-posedness results in certain Sobolev spaces without imposing any restriction on the sign of the potential—whether attractive or repulsive. This study serves as a natural extension of the results presented in (Miao et al., Acta Math. Sci. 42, 2230–2256 (2022)), by addressing the inhomogeneous setting and exploring the problem in higher spatial dimensions. The analysis is particularly challenging due to the simultaneous presence of two complicating factors. First, the inclusion of a Coulomb potential disrupts the scale invariance that is a defining feature of the classical, homogeneous NLS. Second, the equation incorporates a singular, decaying, inhomogeneous term that breaks the space translation invariance typically satisfied by the standard NLS. These two departures from the classical framework introduce significant analytical difficulties, which must be addressed in order to develop a coherent theoretical understanding. To the best of our knowledge, this note represents the first attempt to rigorously study such a formulation of the inhomogeneous NLS in the presence of a Coulomb potential. Furthermore, we mention that this work constitutes an initial step in a broader research program. In an ongoing project, we aim to investigate the long-time, asymptotic behavior of energy-class solutions to this equation, which will further illuminate the qualitative properties of the dynamics under these complex influences.