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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2110.13378 |
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Table of 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 $λ\rightarrow0$.