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| Natura: | Preprint |
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2021
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| Accesso online: | https://arxiv.org/abs/2110.13378 |
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| _version_ | 1866912230002393088 |
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| author | Zhang, Yibin |
| author_facet | Zhang, Yibin |
| contents | Given a smooth bounded domain $Ω$ in $\mathbb{R}^2$, we study the following anisotropic Neumann problem $$ \begin{cases} -\nabla(a(x)\nabla u)+a(x)u=λa(x) u^{p-1}e^{u^p},\,\,\,\, u>0\,\,\,\,\, \textrm{in}\,\,\,\,\, Ω,\\[2mm] \frac{\partial u}{\partialν}=0\,\, \qquad\quad\qquad\qquad\qquad\qquad\qquad \ \ \ \ \,\qquad\quad\, \textrm{on}\,\,\, \partialΩ, \end{cases} $$ where $λ>0$ is a small parameter, $0<p<2$, $a(x)$ is a positive smooth function over $\overlineΩ$ and $ν$ denotes the outer unit normal vector to $\partialΩ$. Under suitable assumptions on anisotropic coefficient $a(x)$, we construct solutions of this problem with arbitrarily many mixed interior and boundary bubbles which concentrate at totally different strict local maximum or minimal boundary points of $a(x)$ restricted to $\partialΩ$, or accumulate to the same strict local maximum boundary point of $a(x)$ over $\overlineΩ$ as $λ\rightarrow0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_13378 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Boundary concentration phenomena for an anisotropic Neumann problem in $\mathbb{R}^2$ Zhang, Yibin Analysis of PDEs Given a smooth bounded domain $Ω$ in $\mathbb{R}^2$, we study the following anisotropic Neumann problem $$ \begin{cases} -\nabla(a(x)\nabla u)+a(x)u=λa(x) u^{p-1}e^{u^p},\,\,\,\, u>0\,\,\,\,\, \textrm{in}\,\,\,\,\, Ω,\\[2mm] \frac{\partial u}{\partialν}=0\,\, \qquad\quad\qquad\qquad\qquad\qquad\qquad \ \ \ \ \,\qquad\quad\, \textrm{on}\,\,\, \partialΩ, \end{cases} $$ where $λ>0$ is a small parameter, $0<p<2$, $a(x)$ is a positive smooth function over $\overlineΩ$ and $ν$ denotes the outer unit normal vector to $\partialΩ$. Under suitable assumptions on anisotropic coefficient $a(x)$, we construct solutions of this problem with arbitrarily many mixed interior and boundary bubbles which concentrate at totally different strict local maximum or minimal boundary points of $a(x)$ restricted to $\partialΩ$, or accumulate to the same strict local maximum boundary point of $a(x)$ over $\overlineΩ$ as $λ\rightarrow0$. |
| title | Boundary concentration phenomena for an anisotropic Neumann problem in $\mathbb{R}^2$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2110.13378 |