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Dependence on parameters of solutions for a generalized poly-Laplacian system on weighted graphs

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Xiaoyu Wang, Junping Xie, Xingyong Zhang, Xin Ou

Analysis of PDEs · Submitted 26 Jun 2024

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Then we mainly show that the mountain pass type nontrivial solutions of the poly-Laplacian system are uniformly bounded for parameters and the concrete upper and lower bounds are given, and are continuously dependent on parameters.

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arXiv ID2406.18384 (version 1)
Pith ID7AYBQVJQ, Pith's permanent record of this paper and its review
CitationJunping Xie, Xiaoyu Wang, Xingyong Zhang, Xin Ou. "Dependence on parameters of solutions for a generalized poly-Laplacian system on weighted graphs." arXiv:2406.18384 (2024). https://pith.science/paper/7AYBQVJQ
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,972 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.11 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.49 · compares with prior results 0.77 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; .

We mainly investigate the continuous dependence on parameters of nontrivial solutions for a generalized poly-Laplacian system on the weighted finite graph $G=(V, E)$. We firstly present an existence result of mountain pass type nontrivial solutions when the nonlinear term $F$ satisfies the super-$(p, q)$ linear growth condition which is a simple generalization of those results in [28]. Then we mainly show that the mountain pass type nontrivial solutions of the poly-Laplacian system are uniformly bounded for parameters and the concrete upper and lower bounds are given, and are continuously dependent on parameters. Similarly, we also present the existence result, the concrete upper and lower bounds, uniqueness, and dependence on parameters for the locally minimum type nontrivial solutions. Subsequently, we present an example on optimal control as an application of our results. Finally, we give a nonexistence result and some results for the corresponding scalar equation.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 We mainly investigate the continuous dependence on parameters of nontrivial solutions for a generalized poly-Laplacian system on the weighted finite graph $G=(V, E)$. contribution 0.96 0.87 0.21 asserted 0.89
s1 We firstly present an existence result of mountain pass type nontrivial solutions when the nonlinear term $F$ satisfies the super-$(p, q)$ linear growth condition which is a simple generalization of those results in [28]. result 0.70 0.89 0.07 asserted 1.00
s2 Then we mainly show that the mountain pass type nontrivial solutions of the poly-Laplacian system are uniformly bounded for parameters and the concrete upper and lower bounds are given, and are continuously dependent on parameters. result 0.91 0.96 0.68 asserted 1.00
s3 Similarly, we also present the existence result, the concrete upper and lower bounds, uniqueness, and dependence on parameters for the locally minimum type nontrivial solutions. result 0.62 0.95 0.13 asserted 1.00
s4 Subsequently, we present an example on optimal control as an application of our results. contribution 0.87 0.92 0.04 asserted 1.00
s5 Finally, we give a nonexistence result and some results for the corresponding scalar equation. result 0.91 0.93 0.05 asserted 0.98

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Original abstract

We mainly investigate the continuous dependence on parameters of nontrivial solutions for a generalized poly-Laplacian system on the weighted finite graph $G=(V, E)$. We firstly present an existence result of mountain pass type nontrivial solutions when the nonlinear term $F$ satisfies the super-$(p, q)$ linear growth condition which is a simple generalization of those results in [28]. Then we mainly show that the mountain pass type nontrivial solutions of the poly-Laplacian system are uniformly bounded for parameters and the concrete upper and lower bounds are given, and are continuously dependent on parameters. Similarly, we also present the existence result, the concrete upper and lower bounds, uniqueness, and dependence on parameters for the locally minimum type nontrivial solutions. Subsequently, we present an example on optimal control as an application of our results. Finally, we give a nonexistence result and some results for the corresponding scalar equation.

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