Uniqueness of positive radial solutions of the Brezis-Nirenberg problem on thin annular domains on $ {\mathbb S}^n $ and symmetry breaking bifurcations

Naoki Shioji; Kohtaro Watanabe

doi:10.3934/cpaa.2020210

Article Contents

2020, Volume 19, Issue 10: 4727-4770. Doi: 10.3934/cpaa.2020210

This issue Previous Article Next Article On some elliptic equation in the whole euclidean space $ \mathbb{R}^2 $ with nonlinearities having new exponential growth condition

Uniqueness of positive radial solutions of the Brezis-Nirenberg problem on thin annular domains on $ {\mathbb S}^n $ and symmetry breaking bifurcations

Naoki Shioji^1, and
Kohtaro Watanabe^2, ,

1.
Department of Mathematics, Faculty of Engineering, Yokohama National University, Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan
2.
Department of Computer Science, National Defense Academy, 1-10-20 Hashirimizu, Yokosuka 239-8686, Japan

^*Corresponding author
^*Corresponding author

Received: October 2018

Revised: May 2020

Published online: October 2020

This work is partially supported by the Grant-in-Aid for Scientific Research (C) (No. 26400160 and No. 18K03387) from Japan Society for the Promotion of Science.

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Abstract

We consider the Brezis-Nirenberg problem

$ \begin{cases} \Delta_{{\mathbb S}^n}U +\lambda U + U^p = 0, \, U>0 & \text{in $\Omega_{\theta_1, \theta_2}$, }\\ U = 0&\text{on $\partial \Omega_{\theta_1, \theta_2}$, } \end{cases} $

where $ \Omega_{\theta_1, \theta_2} $ is the set of the points whose great circle distance from $ (0, \ldots, 0, 1) $ is greater than $ \theta_2 $ and less than $ \theta_1 $. If the annular domain is sufficiently thin, we show that the problem has a unique positive solution whose value depends only on the great circle distance from $ (0, \ldots, 0, 1) $ and there exists a nonradial bifurcation arising from the solution.

Keywords:

Mathematics Subject Classification: 35B32, 35J60, 58J32.

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