  We consider the Cauchy problem for nonlinear Schr\"odinger equations in a
general domain $\Omega\subset\mathbb{R}^N$. 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.
