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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2403.06784 |
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| _version_ | 1866908303215296512 |
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| author | Deng, Haiyun Ji, Jingwen Jiang, Feida Yin, Jiabin |
| author_facet | Deng, Haiyun Ji, Jingwen Jiang, Feida Yin, Jiabin |
| contents | In this paper, we provide an affirmative answer to the {\it conjecture A} for bounded simple rotationally symmetric domains $Ω\subset \mathbb{R}^n(n\geq 3)$ along $x_n$ axis. Precisely, we use a new simple argument to study the symmetry of positive stable solutions for two kinds of semilinear elliptic equations. To do this, when $f(\cdot,s)$ is convex with respect to $s$, we show that the positivity of the first eigenvalue of the corresponding linearized operator in somehow symmetric domains is a sufficient condition for the symmetry of $u$. Moreover, we prove the uniqueness of critical points of a positive stable solution to semilinear elliptic equation $-\triangle u=f(\cdot,u)$ with zero Dirichlet boundary condition for simple rotationally symmetric domains in $\mathbb{R}^n$ by continuity method and a variety of maximum principles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_06784 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniqueness of the critical points of solutions to two kinds of semilinear elliptic equations in higher dimensional domains Deng, Haiyun Ji, Jingwen Jiang, Feida Yin, Jiabin Analysis of PDEs 35B38, 35J05, 35J25 In this paper, we provide an affirmative answer to the {\it conjecture A} for bounded simple rotationally symmetric domains $Ω\subset \mathbb{R}^n(n\geq 3)$ along $x_n$ axis. Precisely, we use a new simple argument to study the symmetry of positive stable solutions for two kinds of semilinear elliptic equations. To do this, when $f(\cdot,s)$ is convex with respect to $s$, we show that the positivity of the first eigenvalue of the corresponding linearized operator in somehow symmetric domains is a sufficient condition for the symmetry of $u$. Moreover, we prove the uniqueness of critical points of a positive stable solution to semilinear elliptic equation $-\triangle u=f(\cdot,u)$ with zero Dirichlet boundary condition for simple rotationally symmetric domains in $\mathbb{R}^n$ by continuity method and a variety of maximum principles. |
| title | Uniqueness of the critical points of solutions to two kinds of semilinear elliptic equations in higher dimensional domains |
| topic | Analysis of PDEs 35B38, 35J05, 35J25 |
| url | https://arxiv.org/abs/2403.06784 |