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On critical and supercritical pseudo-relativistic nonlinear Schrödinger equations

Part of: Equations of mathematical physics and other areas of application Elliptic equations and systems

Published online by Cambridge University Press:  30 January 2019

Woocheol Choi ,
Younghun Hong  and
Jinmyoung Seok
Woocheol Choi
Affiliation:
Department of Mathematics Education, Incheon National University, Incheon22012, Republic of Korea ([email protected])
Younghun Hong
Affiliation:
Department of Mathematics, Chung-Ang University, Seoul06974, Republic of Korea ([email protected])
Jinmyoung Seok
Affiliation:
Department of Mathematics, Kyonggi University, Suwon16227, Republic of Korea ([email protected])
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Abstract

In this paper, we investigate existence and non-existence of a nontrivial solution to the pseudo-relativistic nonlinear Schrödinger equation

$$\left( \sqrt{-c^2\Delta + m^2 c^4}-mc^2\right) u + \mu u = \vert u \vert^{p-1}u\quad {\rm in}~{\open R}^n~(n \ges 2) $$
involving an H1/2-critical/supercritical power-type nonlinearity, that is, p ⩾ ((n + 1)/(n − 1)). We prove that in the non-relativistic regime, there exists a nontrivial solution provided that the nonlinearity is H1/2-critical/supercritical but it is H1-subcritical. On the other hand, we also show that there is no nontrivial bounded solution either (i) if the nonlinearity is H1/2-critical/supercritical in the ultra-relativistic regime or (ii) if the nonlinearity is H1-critical/supercritical in all cases.

Keywords

pseudo-relativistic NLSsupercriticalityexistencenon-existence

MSC classification

Primary: 35J10: Schrödinger operator
Secondary: 35J61: Semilinear elliptic equations 35Q51: Soliton-like equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 3 , June 2020 , pp. 1241 - 1263
Copyright
Copyright © 2019 The Royal Society of Edinburgh

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