Uniqueness and multiple existence of positive radial solutions of the Brezis-Nirenberg Problem on annular domains in ${\Bbb S}^{3}$
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915073195245568 |
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| author | Shioji, Naoki Tanaka, Satoshi Watanabe, Kohtaro |
| author_facet | Shioji, Naoki Tanaka, Satoshi Watanabe, Kohtaro |
| contents | The uniqueness and multiple existence of positive radial solutions to the Brezis-Nirenberg problem on a domain in the 3-dimensional unit sphere ${\mathbb S}^3$ \begin{equation*} \left\{ \begin{aligned} Δ_{{\mathbb S}^3}U -λU + U^p&=0,\, U>0 && \text{in $Ω_{θ_1,θ_2}$,}\\ U &= 0&&\text{on $\partial Ω_{θ_1,θ_2}$,}
\end{aligned} \right. \end{equation*} for $-λ_{1}<λ\leq 1$ are shown, where $Δ_{{\mathbb S}^3}$ is the Laplace-Beltrami operator, $λ_{1}$ is the first eigenvalue of $-Δ_{{\mathbb S}^3}$ and $Ω_{θ_1,θ_2}$ is an annular domain in ${\mathbb S}^3$: whose great circle distance (geodesic distance) from $(0,0,0,1)$ is greater than $θ_1$ and less than $θ_2$. A solution is said to be radial if it depends only on this geodesic distance. It is proved that the number of positive radial solutions of the problem changes with respect to the exponent $p$ and parameter $λ$ when $θ_1=\varepsilon$, $θ_2=π-\varepsilon$ and $0<\varepsilon$ is sufficiently small. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15680 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniqueness and multiple existence of positive radial solutions of the Brezis-Nirenberg Problem on annular domains in ${\Bbb S}^{3}$ Shioji, Naoki Tanaka, Satoshi Watanabe, Kohtaro Analysis of PDEs 34B15, 35J65 The uniqueness and multiple existence of positive radial solutions to the Brezis-Nirenberg problem on a domain in the 3-dimensional unit sphere ${\mathbb S}^3$ \begin{equation*} \left\{ \begin{aligned} Δ_{{\mathbb S}^3}U -λU + U^p&=0,\, U>0 && \text{in $Ω_{θ_1,θ_2}$,}\\ U &= 0&&\text{on $\partial Ω_{θ_1,θ_2}$,} \end{aligned} \right. \end{equation*} for $-λ_{1}<λ\leq 1$ are shown, where $Δ_{{\mathbb S}^3}$ is the Laplace-Beltrami operator, $λ_{1}$ is the first eigenvalue of $-Δ_{{\mathbb S}^3}$ and $Ω_{θ_1,θ_2}$ is an annular domain in ${\mathbb S}^3$: whose great circle distance (geodesic distance) from $(0,0,0,1)$ is greater than $θ_1$ and less than $θ_2$. A solution is said to be radial if it depends only on this geodesic distance. It is proved that the number of positive radial solutions of the problem changes with respect to the exponent $p$ and parameter $λ$ when $θ_1=\varepsilon$, $θ_2=π-\varepsilon$ and $0<\varepsilon$ is sufficiently small. |
| title | Uniqueness and multiple existence of positive radial solutions of the Brezis-Nirenberg Problem on annular domains in ${\Bbb S}^{3}$ |
| topic | Analysis of PDEs 34B15, 35J65 |
| url | https://arxiv.org/abs/2412.15680 |