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Nonnegative solutions of a coupled k-Hessian system involving different fractional Laplacians

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Abstract

This paper studies the following coupled k-Hessian system with different order fractional Laplacian operators:

$$\begin{aligned} {\left\{ \begin{array}{ll} {S_k}({D^2}w(x))-A(x)(-\varDelta )^{\alpha /2}w(x)=f(z(x)),\\ {S_k}({D^2}z(x))-B(x)(-\varDelta )^{\beta /2}z(x)=g(w(x)). \end{array}\right. } \end{aligned}$$

Firstly, we discuss decay at infinity principle and narrow region principle for the k-Hessian system involving fractional order Laplacian operators. Then, by exploiting the direct method of moving planes, the radial symmetry and monotonicity of the nonnegative solutions to the coupled k-Hessian system are proved in a unit ball and the whole space, respectively. We believe that the present work will lead to a deep understanding of the coupled k-Hessian system involving different order fractional Laplacian operators.

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Acknowledgements

Lihong Zhang is supported by National Natural Science Foundation of China (No. 12001344) and Guotao Wang is supported by Natural Science Foundation of Shanxi Province, China (No. 20210302123339).

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Authors and Affiliations

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

    Lihong Zhang, Qi Liu & Guotao Wang

  2. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, 21589, Jeddah, Saudi Arabia

    Bashir Ahmad

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  1. Lihong Zhang
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Zhang, L., Liu, Q., Ahmad, B. et al. Nonnegative solutions of a coupled k-Hessian system involving different fractional Laplacians. Fract Calc Appl Anal (2024). https://doi.org/10.1007/s13540-024-00277-1

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