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| Format: | Preprint |
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2021
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| Online Access: | https://arxiv.org/abs/2112.01755 |
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| _version_ | 1866912159999459328 |
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| author | Hou, Yongjun Pinchover, Yehuda Rasila, Antti |
| author_facet | Hou, Yongjun Pinchover, Yehuda Rasila, Antti |
| contents | In this paper, we study positive solutions of the quasilinear elliptic equation $$Q'_{p,\mathcal{A},V}[u]\triangleq-\mathrm{div}{\mathcal{A}(x,\nabla u)}+V(x)|u|^{p-2}u=0,$$ in a domain $Ω\subseteq \mathbb{R}^n$, where $n\geq 2$, $1<p<\infty$, the divergence of $\mathcal{A}$ is the well known $\mathcal{A}$-Laplace operator considered in the influential book of Heinonen, Kilpeläinen, and Martio, and the potential $V$ belongs to a certain local Morrey space. The main aim of the paper is to extend criticality theory to the operator $Q'_{p,\mathcal{A},V}$. In particular, we prove an Agmon-Allegretto-Piepenbrink (AAP) type theorem, establish the uniqueness and simplicity of the principal eigenvalue of $Q'_{p,\mathcal{A},V}$ in a domain $ω\SubsetΩ$, and give various characterizations of criticality. Furthermore, we also study positive solutions of the equation $Q'_{p,\mathcal{A},V}[u]=0$ of minimal growth at infinity in $Ω$, the existence of a minimal positive Green function, and the minimal decay at infinity of Hardy-weights. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_01755 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Positive solutions of the $\mathcal{A}$-Laplace equation with a potential Hou, Yongjun Pinchover, Yehuda Rasila, Antti Analysis of PDEs Primary 35B09, Secondary 35B50, 35J08, 35J62 In this paper, we study positive solutions of the quasilinear elliptic equation $$Q'_{p,\mathcal{A},V}[u]\triangleq-\mathrm{div}{\mathcal{A}(x,\nabla u)}+V(x)|u|^{p-2}u=0,$$ in a domain $Ω\subseteq \mathbb{R}^n$, where $n\geq 2$, $1<p<\infty$, the divergence of $\mathcal{A}$ is the well known $\mathcal{A}$-Laplace operator considered in the influential book of Heinonen, Kilpeläinen, and Martio, and the potential $V$ belongs to a certain local Morrey space. The main aim of the paper is to extend criticality theory to the operator $Q'_{p,\mathcal{A},V}$. In particular, we prove an Agmon-Allegretto-Piepenbrink (AAP) type theorem, establish the uniqueness and simplicity of the principal eigenvalue of $Q'_{p,\mathcal{A},V}$ in a domain $ω\SubsetΩ$, and give various characterizations of criticality. Furthermore, we also study positive solutions of the equation $Q'_{p,\mathcal{A},V}[u]=0$ of minimal growth at infinity in $Ω$, the existence of a minimal positive Green function, and the minimal decay at infinity of Hardy-weights. |
| title | Positive solutions of the $\mathcal{A}$-Laplace equation with a potential |
| topic | Analysis of PDEs Primary 35B09, Secondary 35B50, 35J08, 35J62 |
| url | https://arxiv.org/abs/2112.01755 |