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 · Communications on Pure & Applied Analysis · 2020
We consider the Brezis-Nirenberg problem \begin{document}$ \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} $\end{document} 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.