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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2401.15406 |
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| _version_ | 1866911957664137216 |
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| author | Chata, Juan Carlos Ortiz Petitta, Francesco |
| author_facet | Chata, Juan Carlos Ortiz Petitta, Francesco |
| contents | In this paper we analyze the asymptotic behaviour as $p\to 1^+$ of solutions $u_p$ to $$ \left\{ \begin{array}{rclr} -Δ_p u_p&=&\fracλ{|x|^p}|u_p|^{p-2}u_p+f&\quad \mbox{ in } Ω,\\ u_p&=&0 &\quad \mbox{ on }\partialΩ, \end{array}\right. $$ where $Ω$ is a bounded open subset of $\mathbb{R}^N$ with Lipschitz boundary, $λ\in\mathbb{R}^+$, and $f$ is a nonnegative datum in $L^{N,\infty}(Ω)$. Under sharp smallness assumptions on the data $λ$ and $f$ we prove that $u_p$ converges to a suitable solution to the homogeneous Dirichlet problem $$\left\{ \begin{array}{rclr}- Δ_{1} u &=& \fracλ{|x|}{\rm Sgn}(u)+f & \text{in}\, Ω,\\ u&=&0 & \text{on}\ \partial Ω,\end{array}\right. $$ where $Δ_{1} u ={\rm div}\left(\frac{D u}{|Du|}\right)$ is the $1$-Laplace operator. The main assumptions are further discussed through explicit examples in order to show their optimality. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_15406 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence, non-existence and degeneracy of limit solutions to $p-$Laplace problems involving Hardy potentials as $p\to1^+$ Chata, Juan Carlos Ortiz Petitta, Francesco Analysis of PDEs In this paper we analyze the asymptotic behaviour as $p\to 1^+$ of solutions $u_p$ to $$ \left\{ \begin{array}{rclr} -Δ_p u_p&=&\fracλ{|x|^p}|u_p|^{p-2}u_p+f&\quad \mbox{ in } Ω,\\ u_p&=&0 &\quad \mbox{ on }\partialΩ, \end{array}\right. $$ where $Ω$ is a bounded open subset of $\mathbb{R}^N$ with Lipschitz boundary, $λ\in\mathbb{R}^+$, and $f$ is a nonnegative datum in $L^{N,\infty}(Ω)$. Under sharp smallness assumptions on the data $λ$ and $f$ we prove that $u_p$ converges to a suitable solution to the homogeneous Dirichlet problem $$\left\{ \begin{array}{rclr}- Δ_{1} u &=& \fracλ{|x|}{\rm Sgn}(u)+f & \text{in}\, Ω,\\ u&=&0 & \text{on}\ \partial Ω,\end{array}\right. $$ where $Δ_{1} u ={\rm div}\left(\frac{D u}{|Du|}\right)$ is the $1$-Laplace operator. The main assumptions are further discussed through explicit examples in order to show their optimality. |
| title | Existence, non-existence and degeneracy of limit solutions to $p-$Laplace problems involving Hardy potentials as $p\to1^+$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2401.15406 |