Construction of two-bubble solutions for the energy-critical NLS

Jendrej, Jacek

doi:10.2140/apde.2017.10.1923

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Construction of two-bubble solutions for the energy-critical NLS

Jacek Jendrej

Vol. 10 (2017), No. 8, 1923–1959

DOI: 10.2140/apde.2017.10.1923

Abstract

We construct pure two-bubbles for the energy-critical focusing nonlinear Schrödinger equation in space dimension $N \geq 7$ . The constructed solution is global in (at least) one time direction and approaches a superposition of two stationary states both centered at the origin, with the ratio of their length scales converging to $0$ . One of the bubbles develops at scale $1$ , whereas the length scale of the other converges to $0$ at rate $| t |^{- \frac{2}{N - 6}}$ . The phases of the two bubbles form the right angle.

Keywords

nonlinear Schrödinger equation, energy-critical, multisoliton

Mathematical Subject Classification 2010

Primary: 35B40

Milestones

Received: 7 November 2016

Revised: 8 May 2017

Accepted: 28 July 2017

Published: 18 August 2017

Authors

Jacek Jendrej
Department of Mathematics University of Chicago Chicago, IL % 60615 United States

Vol. 10, No. 8, 2017