Existence results for variational quasilinear elliptic systems involving the vectorial $p$-Laplacian
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910062237188096 |
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| author | Canino, Annamaria Mauro, Simone |
| author_facet | Canino, Annamaria Mauro, Simone |
| contents | We prove existence and regularity results for the following elliptic system: \[ \begin{cases} -\textbf{div}(|D\boldsymbol{u}|^{p-2}D\boldsymbol{u})=\boldsymbol{f}(x,\boldsymbol{u}) & \text{in } Ω\\ \boldsymbol{u}=0 & \text{on } \partialΩ, \end{cases} \] where $\boldsymbol{u}=(u^1,\dots,u^m)$, $p>1$, and $Ω\subset\mathbb{R}^N$ is a bounded domain. We also consider the special case
\[\boldsymbol{f}(x,\boldsymbol{u})=λ|\boldsymbol{u}|^{p-2}\boldsymbol{u}+|\boldsymbol{u}|^{q-2}\boldsymbol{u},\] and we prove a classification result. In particular, we show that any least energy solution is of the form $(c^1ω,\dots,c^mω)$, where $\boldsymbol{c}=(c^1,\dots,c^m)\in S^{m-1}$ (the $(m-1)$-sphere in $\mathbb R^m$) and $ω$ is a positive solution of the corresponding scalar equation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_15694 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence results for variational quasilinear elliptic systems involving the vectorial $p$-Laplacian Canino, Annamaria Mauro, Simone Analysis of PDEs 35A01, 35A15, 35J05, 35J20, 35J25, 35J62 We prove existence and regularity results for the following elliptic system: \[ \begin{cases} -\textbf{div}(|D\boldsymbol{u}|^{p-2}D\boldsymbol{u})=\boldsymbol{f}(x,\boldsymbol{u}) & \text{in } Ω\\ \boldsymbol{u}=0 & \text{on } \partialΩ, \end{cases} \] where $\boldsymbol{u}=(u^1,\dots,u^m)$, $p>1$, and $Ω\subset\mathbb{R}^N$ is a bounded domain. We also consider the special case \[\boldsymbol{f}(x,\boldsymbol{u})=λ|\boldsymbol{u}|^{p-2}\boldsymbol{u}+|\boldsymbol{u}|^{q-2}\boldsymbol{u},\] and we prove a classification result. In particular, we show that any least energy solution is of the form $(c^1ω,\dots,c^mω)$, where $\boldsymbol{c}=(c^1,\dots,c^m)\in S^{m-1}$ (the $(m-1)$-sphere in $\mathbb R^m$) and $ω$ is a positive solution of the corresponding scalar equation. |
| title | Existence results for variational quasilinear elliptic systems involving the vectorial $p$-Laplacian |
| topic | Analysis of PDEs 35A01, 35A15, 35J05, 35J20, 35J25, 35J62 |
| url | https://arxiv.org/abs/2510.15694 |