Unconditional finite-time blow-up of the inhomogeneous NLS with Coulomb potential

In this work, we investigate an inhomogeneous nonlinear Schrödinger equation that includes a Coulomb-type potential. Our primary objective is to establish finite-time blow-up of energy solutions. The aim is to avoid imposing any radial or finite-variance assumptions on the datum. The challenge is to overcome the presence of the Coulomb potential, which breaks the scale invariance of the standard nonlinear Schrödinger equation (NLS). The presence of an inhomogeneous decaying source term enables us to avoid any spherical symmetry assumption. We discuss two different cases: The inter-critical regime under the ground-state threshold and the mass-critical regime with negative energy. The method follows a localized variance-type identity coupled with some variational analysis. Since the Coulomb potential does not have the same scaling as the Laplacian operator, we express the threshold by using the ground state of the associated INLS with no potential. Some numerical simulations, which confirm the theoretical result, are given at the end of this paper.