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arXiv · 1712.10239

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

Masayuki Hayashi
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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.

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Masayuki Hayashi. 2017-12-29. A note on the nonlinear Schr\"odinger equation in a general domain. https://doi.org/10.1016/j.na.2018.03.017

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