Abstract
In this work, we explore the asymptotic behaviors of positive solutions to the \(\sigma _k\)-Yamabe equation. Extending Han and Li’s previous work on the Yamabe equation, we demonstrate that for every approximate solution \({\widetilde{w}}\) of a specified order, there exists a corresponding solution w that closely approximates \({\widetilde{w}}\). Our study further presents a concrete method for constructing these approximate solutions. By appropriately perturbing the radial solution, we can consistently obtain an approximate solution with a designated order. This approach offers significant insights into the characteristics and behaviors of solutions near isolated singular points.
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Acknowledgements
The first author acknowledges the support of Xingwang Xu’s start up funds from Nanjing University (No. 020314912205) and Xu’s research grant (NFSC No. 00127205).
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Communicated by Yanyan Li.
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Li, Z., Han, Q. Singular solutions to the \(\sigma _k\)-Yamabe equation with prescribed asymptotics. Calc. Var. 63, 64 (2024). https://doi.org/10.1007/s00526-024-02675-y
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DOI: https://doi.org/10.1007/s00526-024-02675-y