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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.00708 |
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| _version_ | 1866914470256705536 |
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| author | Gou, Tianxiang |
| author_facet | Gou, Tianxiang |
| contents | In this paper, we consider the radial symmetry, uniqueness and non-degeneracy of solutions to the degenerate nonlinear elliptic equation $$ -\nabla \cdot \left(|x|^{2a} \nabla u\right) + ωu=|u|^{p-2}u \quad \mbox{in} \,\, \R^d, $$ where $d \geq 2$, $0<a<1$, $ω>0$ and $2<p<\frac{2d}{d-2(1-a)}$. We proved that any ground state is radially symmetric and strictly decreasing in the radial direction. Moreover, we establish the uniqueness of ground states and derive the non-degeneracy of ground states in the corresponding radially symmetric Sobolev space. This affirms the natural conjectures posed recently in \cite{IS}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_00708 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Radial symmetry, uniqueness and non-degeneracy of solutions to degenerate nonlinear Schrödinger equations Gou, Tianxiang Analysis of PDEs 35Q55, 35B35 In this paper, we consider the radial symmetry, uniqueness and non-degeneracy of solutions to the degenerate nonlinear elliptic equation $$ -\nabla \cdot \left(|x|^{2a} \nabla u\right) + ωu=|u|^{p-2}u \quad \mbox{in} \,\, \R^d, $$ where $d \geq 2$, $0<a<1$, $ω>0$ and $2<p<\frac{2d}{d-2(1-a)}$. We proved that any ground state is radially symmetric and strictly decreasing in the radial direction. Moreover, we establish the uniqueness of ground states and derive the non-degeneracy of ground states in the corresponding radially symmetric Sobolev space. This affirms the natural conjectures posed recently in \cite{IS}. |
| title | Radial symmetry, uniqueness and non-degeneracy of solutions to degenerate nonlinear Schrödinger equations |
| topic | Analysis of PDEs 35Q55, 35B35 |
| url | https://arxiv.org/abs/2503.00708 |