A note on the nonlinear Schrödinger equation in a general domain

Abstract

We consider the Cauchy problem for nonlinear Schrödinger 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.

minor revision; 24 pages. To appear in Nonlinear Analysis

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