A complete description of the asymptotic behavior at infinity of positive radial solutions to $Δ^2 u = u^α$ in $\mathbf R^n$
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2018
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| _version_ | 1866910669531512832 |
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| author | Ngô, Quôc Anh Nguyen, Van Hoang Phan, Quoc Hung |
| author_facet | Ngô, Quôc Anh Nguyen, Van Hoang Phan, Quoc Hung |
| contents | We consider the biharmonic equation $Δ^2 u = u^α$ in $\mathbf R^n$ with $n \geqslant 1$. It was proved that this equation has a positive classical solution if, and only if, either $α\leqslant 1$ with $n \geqslant 1$ or $α\geqslant (n+4)/(n-4)$ with $n \geqslant 5$. The asymptotic behavior at infinity of all positive radial solutions was known in the case $α\geqslant (n+4)/(n-4)$ and $n \geqslant 5$. In this paper, we classify the asymptotic behavior at infinity of all positive radial solutions in the remaining case $α\leqslant 1$ with $n \geqslant 1$; hence obtaining a complete picture of the asymptotic behavior at infinity of positive radial solutions. Since the underlying equation is higher order, we propose a new approach which relies on a representation formula and asymptotic analysis arguments. We believe that the approach introduced here can be conveniently applied to study other problems with higher order operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1803_11520 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | A complete description of the asymptotic behavior at infinity of positive radial solutions to $Δ^2 u = u^α$ in $\mathbf R^n$ Ngô, Quôc Anh Nguyen, Van Hoang Phan, Quoc Hung Analysis of PDEs Primary 35B08, 35B40, 35J91, Secondary 35B09, 35B51 We consider the biharmonic equation $Δ^2 u = u^α$ in $\mathbf R^n$ with $n \geqslant 1$. It was proved that this equation has a positive classical solution if, and only if, either $α\leqslant 1$ with $n \geqslant 1$ or $α\geqslant (n+4)/(n-4)$ with $n \geqslant 5$. The asymptotic behavior at infinity of all positive radial solutions was known in the case $α\geqslant (n+4)/(n-4)$ and $n \geqslant 5$. In this paper, we classify the asymptotic behavior at infinity of all positive radial solutions in the remaining case $α\leqslant 1$ with $n \geqslant 1$; hence obtaining a complete picture of the asymptotic behavior at infinity of positive radial solutions. Since the underlying equation is higher order, we propose a new approach which relies on a representation formula and asymptotic analysis arguments. We believe that the approach introduced here can be conveniently applied to study other problems with higher order operators. |
| title | A complete description of the asymptotic behavior at infinity of positive radial solutions to $Δ^2 u = u^α$ in $\mathbf R^n$ |
| topic | Analysis of PDEs Primary 35B08, 35B40, 35J91, Secondary 35B09, 35B51 |
| url | https://arxiv.org/abs/1803.11520 |