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On supercritical nonlinear Schrödinger equations with ellipse-shaped potentials

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  02 December 2019

Jianfu Yang  and
Jinge Yang
Jianfu Yang
Affiliation:
Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi330022, P. R. China (jfyang_2000@yahoo.com)
Jinge Yang*
Affiliation:
School of Sciences, Nanchang Institute of Technology, Nanchang330099, P. R. China (jgyang2007@yeah.net)
*
*Corresponding author.
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Abstract

In this paper, we study the existence and concentration of normalized solutions to the supercritical nonlinear Schrödinger equation

\[ \left\{\begin{array}{@{}ll} -\Delta u + V(x) u = \mu_q u + a \vert u \vert ^q u & {\rm in}\ \mathbb{R}^2,\\ \int_{\mathbb{R}^2} \vert u \vert ^2\,{\rm d}x =1, & \end{array} \right.\]
where μq is the Lagrange multiplier. For ellipse-shaped potentials V(x), we show that for q > 2 close to 2, the equation admits an excited solution uq, and furthermore, we study the limiting behaviour of uq when q → 2+. Particularly, we describe precisely the blow-up formation of the excited state uq.

Keywords

Nonlinear elliptic equationL2 supercriticalconstrained minimizationGross–Pitaevskii functionalellipse-shaped potential

MSC classification

Primary: 35B38: Critical points
Secondary: 35J20: Variational methods for second-order elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 3187 - 3215
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Copyright
Copyright © 2019 The Royal Society of Edinburgh

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