Abstract
In this article, we consider the Einstein-scalar field Lichnerowicz equation
on any connected finite graph \(G=(V,E)\), where A, B, h are given functions on V with \(A\ge 0\), \(A\not \equiv 0\) on V, and \(p>2\) is a constant. By using the classical variational method, topological degree theory and heat-flow method, we provide a systematical study on this equation by providing the existence results for each case: positive, negative and null Yamabe-scalar field conformal invariant, namely \(h>0\), \(h<0\) and \(h=0\) respectively.
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Acknowledgements
Y. Liu is partially supported by the National Key R &D Program of China 2022YFA1005400 and NSFC No. 11971026, NSFC No. 12141105. C. Wang is partially supported by NSFC No.12071169. J. Wang is partially supported by National Key R &D Program of China 2022YFA1005601 and NSFC No.12371114. W. Yang is partially supported by National Key R &D Program of China 2022YFA1006800, NSFC No.12171456, NSFC No.12271369 and No. SRG2023-00067-FST.
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Innovative Research Group Project of the National Natural Science Foundation of China 11971202
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Cui, L., Liu, Y., Wang, C. et al. The Einstein-scalar field Lichnerowicz equations on graphs. Calc. Var. 63, 138 (2024). https://doi.org/10.1007/s00526-024-02737-1
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DOI: https://doi.org/10.1007/s00526-024-02737-1