A note on the nonlinear Schr\"odinger equation in a general domain

TL;DR

This paper presents a new, simplified method for constructing solutions to the nonlinear Schrödinger equation in general domains, using a Cauchy sequence approach instead of classical compactness methods.

Contribution

It introduces an alternative approach to solution construction for nonlinear Schrödinger equations, applicable to various nonlinearities in arbitrary domains.

Findings

Solutions form a Cauchy sequence in a Banach space

Applicable to power, logarithmic, and damping nonlinearities

Simplifies the existence proof process

Abstract

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.