Stationary states for non linear one-dimensional Schrِdinger equations with singular potential

In this paper we consider the time-independent one-dimensional non linear Schrödinger equation (NLS) with pointwise singular potential. We prove that when the strength of the pointwise interaction is less than a critical value, depending on the nonlinearity power σ , then a non linear real-valued bound state exists. Furthermore, we show that when σ is larger than 2 a further new real-valued stationary state appears under some conditions.