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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2401.10485 |
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| _version_ | 1866913555149750272 |
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| author | Ren, Qiang |
| author_facet | Ren, Qiang |
| contents | We consider the following anisotropic sinh-Poisson tpye equation with a Hardy or Hénon term:
$$-\mathrm{div} (a(x)\nabla u)+ a(x)u=\varepsilon^2a(x)|x-q|^{2α}(e^u-e^{-u}) \quad\mathrm{in}\quad Ω,$$ $$\frac{\partial u}{\partial n}=0,\quad \mathrm{on}\quad \partialΩ,$$ where $\varepsilon>0$, $q\in \barΩ\subset \mathbb{R}^2$, $α\in(-1,\infty)- \mathbb{N}$, $Ω\subset \mathbb{R}^2$ is a smooth bounded domain, $n$ is the unit outward normal vector of $\partial Ω$ and $a(x)$ is a smooth positive function defined on $\barΩ$. From finite dimensional reduction method, we proved that this problem has a sequence of sign-changing solutions with arbitrarily many interior spikes accumulating to $q$, provided $q\in Ω$ is a local maximizer of $a(x)$. However, if $q\in \partial Ω$ is a strict local maximum point of $a(x)$ and satisfies $\langle \nabla a(q),n \rangle=0$, we proved that this problem has a family of sign-changing solutions with arbitrarily many mixed interior and boundary spikes accumulating to $q$.
Under the same condition, we could also construct a sequence of blow-up solutions for the following problem $$ -\mathrm{div} (a(x)\nabla u)+ a(x)u=\varepsilon^2a(x)|x-q|^{2α}e^u\quad \mathrm{in} \quadΩ,$$ $$\frac{\partial u}{\partial n}=0, \quad \mathrm{on}\quad \partialΩ.$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_10485 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sign-changing concentration phenomena of an anisotropic sinh-Poisson type equation with a Hardy or Hénon term Ren, Qiang Analysis of PDEs We consider the following anisotropic sinh-Poisson tpye equation with a Hardy or Hénon term: $$-\mathrm{div} (a(x)\nabla u)+ a(x)u=\varepsilon^2a(x)|x-q|^{2α}(e^u-e^{-u}) \quad\mathrm{in}\quad Ω,$$ $$\frac{\partial u}{\partial n}=0,\quad \mathrm{on}\quad \partialΩ,$$ where $\varepsilon>0$, $q\in \barΩ\subset \mathbb{R}^2$, $α\in(-1,\infty)- \mathbb{N}$, $Ω\subset \mathbb{R}^2$ is a smooth bounded domain, $n$ is the unit outward normal vector of $\partial Ω$ and $a(x)$ is a smooth positive function defined on $\barΩ$. From finite dimensional reduction method, we proved that this problem has a sequence of sign-changing solutions with arbitrarily many interior spikes accumulating to $q$, provided $q\in Ω$ is a local maximizer of $a(x)$. However, if $q\in \partial Ω$ is a strict local maximum point of $a(x)$ and satisfies $\langle \nabla a(q),n \rangle=0$, we proved that this problem has a family of sign-changing solutions with arbitrarily many mixed interior and boundary spikes accumulating to $q$. Under the same condition, we could also construct a sequence of blow-up solutions for the following problem $$ -\mathrm{div} (a(x)\nabla u)+ a(x)u=\varepsilon^2a(x)|x-q|^{2α}e^u\quad \mathrm{in} \quadΩ,$$ $$\frac{\partial u}{\partial n}=0, \quad \mathrm{on}\quad \partialΩ.$$ |
| title | Sign-changing concentration phenomena of an anisotropic sinh-Poisson type equation with a Hardy or Hénon term |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2401.10485 |