Abstract
In this paper, we study positive solutions of the quasilinear elliptic equation
in a domain \({\Omega }\subseteq \mathbb {R}^{n}\), where n ≥ 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 \(Q^{\prime }_{p,\mathcal {A},V}\). In particular, we prove an Agmon-Allegretto-Piepenbrink (AAP) type theorem, establish the uniqueness and simplicity of the principal eigenvalue of \(Q^{\prime }_{p,\mathcal {A},V}\) in a domain \(\omega \Subset {\Omega }\), and give various characterizations of criticality. Furthermore, we also study positive solutions of the equation \(Q^{\prime }_{p,\mathcal {A},V}[u]=0\) of minimal growth at infinity in Ω, the existence of a minimal positive Green function, and the minimal decay at infinity of Hardy-weights.
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This paper is based on the thesis of the first author for the M.Sc. degree in Mathematics conferred by the Technion-Israel Institute of Technology under the supervision of Professors Yehuda Pinchover and Antti Rasila and on his subsequent Ph.D. research at the Technion. Y.H. and A.R. gratefully acknowledge the generous financial help of NNSF of China (No. 11971124) and NSF of Guangdong Province (No. 2021A1515010326). Y.H. and Y.P. acknowledge the support of the Israel Science Foundation (grant 637/19) founded by the Israel Academy of Sciences and Humanities. Y.H. is grateful to the Technion for supporting his study.
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Hou, Y., Pinchover, Y. & Rasila, A. Positive Solutions of the \(\mathcal {A}\)-Laplace Equation with a Potential. Potential Anal 60, 721–758 (2024). https://doi.org/10.1007/s11118-023-10068-7
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DOI: https://doi.org/10.1007/s11118-023-10068-7
Keywords
- Agmon-Allegretto-Piepenbrink theorem
- \(\mathcal {A}\)-Laplacian
- Criticality theory
- Positive solutions
- Principal eigenvalue
- Minimal growth