Sign-changing solution for an overdetermined elliptic problem on unbounded domain
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916477228023808 |
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| author | Dai, Guowei Zhang, Yong |
| author_facet | Dai, Guowei Zhang, Yong |
| contents | We prove the existence of two smooth families of unbounded domains in $\mathbb{R}^{N+1}$ with $N\geq1$ such that \begin{equation} -Δu=λu\,\, \text{in}\,\,Ω, \,\, u=0,\,\,\partial_νu=\text{const}\,\,\text{on}\,\,\partialΩ\nonumber \end{equation} admits a sign-changing solution. The domains bifurcate from the straight cylinder $B_1\times \mathbb{R}$, where $B_1$ is the unit ball in $\mathbb{R}^N$. These results can be regarded as counterexamples to the Berenstein conjecture on unbounded domain. Unlike most previous papers in this direction, a very delicate issue here is that there may be two-dimensional kernel space at some bifurcation point. Thus a Crandall-Rabinowitz type bifurcation theorem from high-dimensional kernel space is also established to achieve the goal. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_05550 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sign-changing solution for an overdetermined elliptic problem on unbounded domain Dai, Guowei Zhang, Yong Analysis of PDEs We prove the existence of two smooth families of unbounded domains in $\mathbb{R}^{N+1}$ with $N\geq1$ such that \begin{equation} -Δu=λu\,\, \text{in}\,\,Ω, \,\, u=0,\,\,\partial_νu=\text{const}\,\,\text{on}\,\,\partialΩ\nonumber \end{equation} admits a sign-changing solution. The domains bifurcate from the straight cylinder $B_1\times \mathbb{R}$, where $B_1$ is the unit ball in $\mathbb{R}^N$. These results can be regarded as counterexamples to the Berenstein conjecture on unbounded domain. Unlike most previous papers in this direction, a very delicate issue here is that there may be two-dimensional kernel space at some bifurcation point. Thus a Crandall-Rabinowitz type bifurcation theorem from high-dimensional kernel space is also established to achieve the goal. |
| title | Sign-changing solution for an overdetermined elliptic problem on unbounded domain |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2304.05550 |