| Abstract: |
| In this talk, we consider the Dirichlet problem: $\Delta_{\mathbb{S}^3} u - u + u^p=0$ in $\Omega_\varepsilon$; $u=0$ on $\partial\Omega_\varepsilon$, where $\Delta_{\mathbb{S}^3}$ is the Laplace-Beltrami operator on the three-dimensional unit sphere $\mathbb{S}^3$, $p>1$, $0< \varepsilon < \pi/2$, and $\Omega_\varepsilon$ is an annular domain in $\mathbb{S}^3$ whose great circle distance (geodesic distance) from the North Pole is greater than $\varepsilon$ and less than $\pi-\varepsilon$. We obtain the existence, uniqueness, and multiplicity results of the positive solutions depend only on the latitude. This is joint work with Naoki Shioji (Yokohama National University) and Kohtaro Watanabe (National Defense Academy). |
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