Trapping by repulsion: The NLS with a delta-prime

We establish the existence and provide explicit expressions for the stationary states of the one-dimensional Schr¨odinger equation with a repulsive delta-prime potential and a focusing nonlinearity of power type. Furthermore, we prove that, if the nonlinearity is subcritical, then ground states exist for any strength of the delta-prime interaction and for every positive value of the mass. This result supplies an example of ground states arising from a repulsive potential, a counterintuitive phenomenon explained by the emergence of an additional dimension in the energy space, induced by the delta-prime interaction. This new dimension contains states of lower energy and is thus responsible for the existence of nonlinear ground states that do not originate from linear eigenfunctions. The explicit form of the ground states is derived by addressing the related problem of minimizing the action functional on the Nehari manifold. We solve such problem in the subcritical, critical, and supercritical regimes