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Complex-valued Solutions of the Planar Schr\'odinger-Newton System
- Publication Year :
- 2023
-
Abstract
- In this paper, we consider complex-valued solutions of the planar Schr\"odinger-Newton system, which can be described by minimizers of the constraint minimization problem. It is shown that there exists a critical rotational velocity $0<\Omega^*\leq \infty$, depending on the general trapping potential $V(x)$, such that for any rotational velocity $0\leq\Omega<\Omega^*$, minimizers exist if and only if $0<a<a^*:=\|Q\|_{2}^{2}$, where $Q>0$ is the unique positive solution of $-\Delta u+u-u^3=0$ in $\mathbb{R}^2$. Moreover, under some suitable assumptions on $V(x)$, applying blow-up analysis and energy estimates, we present a detailed analysis on the concentration behavior of minimizers as $a\nearrow a^*$.<br />Comment: arXiv admin note: text overlap with arXiv:2212.00234 by other authors
- Subjects :
- Mathematics - Analysis of PDEs
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2304.11308
- Document Type :
- Working Paper