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The multiplicity of radial p-k-convex solutions for the p-k-Hessian equation

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Abstract

This paper focuses on radial p-k-convex solutions for the following p-k-Hessian equation

$$\begin{aligned} \left\{ \begin{array}{ll} S_{k}(\xi (D_{i}(|Dv|^{p-2}D_{j}v)))=M(|z|){(|v|+1)}^{m}(ln (|v|+1))^{\mu }, z\in E,\\ v=+\infty , ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~z\in \partial E, \end{array} \right. \end{aligned}$$

where \(p\ge 2\), \(k\in \{1,2,...,n\}\), \(E\subset \mathbb {R}^{n}(n\ge 2)\) denotes a ball. For the case of \(0<m<(p-1)k\), \(\mu =0\), the multiplicity of radial p-k-convex solutions of the above p-k-Hessian equation is established by the sub-supersolutions method. For the case of \(m=(p-1)k\), \(\mu >(p-1)k\), we construct a new supporting function to overcome the difficulty caused by logarithmic nonlinearity, which ensures that the above p-k-Hessian equation has infinitely many radial p-k-convex solutions.

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Authors and Affiliations

  1. School of Mathematics and Computer Science, Shanxi Normal University, Taiyuan, 030031, Shanxi, China

    Guotao Wang & Mengjie Guo

  2. Department of Technical Sciences, Western Caspian University, Baku, Azerbaijan

    Guotao Wang

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  1. Guotao Wang
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  2. Mengjie Guo
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Correspondence to Guotao Wang.

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The work is supported by Natural Science Foundation of Shanxi, China (No. 20210302123339), the Graduate Education Innovation Program of Shanxi, China (No. 2024JG103) and Postgraduate Education Innovation Program of Shanxi Normal University, China(No. 2024YJSKCSZSFK-06).

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Wang, G., Guo, M. The multiplicity of radial p-k-convex solutions for the p-k-Hessian equation. J. Appl. Math. Comput. 71, 927–943 (2025). https://doi.org/10.1007/s12190-024-02262-6

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  • DOI: https://doi.org/10.1007/s12190-024-02262-6

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