A Hardy-H{é}non equation in $\mathbb{R}^N$ with sublinear absorption
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arXiv
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| Format: | Preprint |
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2024
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| author | Iagar, Razvan Gabriel Laurençot, Philippe |
| author_facet | Iagar, Razvan Gabriel Laurençot, Philippe |
| contents | Consider $m\>1$, $N\ge 1$ and $\max\{-2,-N\}\<σ\<0$. The Hardy-Hénon equation with sublinear absorption\begin(equation*}- Δv(x) - |x|^σv(x) + \frac{1}{m-1} v^{1/m}(x)= 0, \qquad x\in\mathbb{R}^N,\end{equation*}is shown to have at least one solution $v\in H^1(\mathbb{R}^N)\cap L^{(m+1)/m}(\mathbb{R}^N)$, which is non-negative and radially symmetric with a non-increasing profile. In addition, any such solution is compactly supported, bounded and enjoys the better regularity $v\in W^{2,q}(\mathbb{R}^N)$ for $q\in [1,N/|σ|)$. A key ingredient in the proof is a particular case of the celebrated Caffarelli-Kohn-Nirenberg inequalities, for which we obtain the existence of an extremal function which is non-negative, bounded, compactly supported and radially symmetric with a non-increasing profile.A by-product of these results is the existence of compactly supported separate variables solutions to a porous medium equation with a spatially dependent source featuring a singular coefficient. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_05909 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Hardy-H{é}non equation in $\mathbb{R}^N$ with sublinear absorption Iagar, Razvan Gabriel Laurençot, Philippe Analysis of PDEs Consider $m\>1$, $N\ge 1$ and $\max\{-2,-N\}\<σ\<0$. The Hardy-Hénon equation with sublinear absorption\begin(equation*}- Δv(x) - |x|^σv(x) + \frac{1}{m-1} v^{1/m}(x)= 0, \qquad x\in\mathbb{R}^N,\end{equation*}is shown to have at least one solution $v\in H^1(\mathbb{R}^N)\cap L^{(m+1)/m}(\mathbb{R}^N)$, which is non-negative and radially symmetric with a non-increasing profile. In addition, any such solution is compactly supported, bounded and enjoys the better regularity $v\in W^{2,q}(\mathbb{R}^N)$ for $q\in [1,N/|σ|)$. A key ingredient in the proof is a particular case of the celebrated Caffarelli-Kohn-Nirenberg inequalities, for which we obtain the existence of an extremal function which is non-negative, bounded, compactly supported and radially symmetric with a non-increasing profile.A by-product of these results is the existence of compactly supported separate variables solutions to a porous medium equation with a spatially dependent source featuring a singular coefficient. |
| title | A Hardy-H{é}non equation in $\mathbb{R}^N$ with sublinear absorption |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2410.05909 |