A note on the nonlinear Schrödinger equation in a general domain

We consider the Cauchy problem for nonlinear Schrödinger equations in a general domain Ω⊂RN. Construction of solutions has been only done by classical compactness method in previous results. Here, we construct solutions by a simple alternative approach. More precisely, solutions are constructed by proving that approximate solutions form a Cauchy sequence in some Banach space. We discuss three different types of nonlinearities: power type nonlinearities, logarithmic nonlinearities and damping nonlinearities.