The Asymptotic Behavior of Solutions to Quasilinear Elliptic Equation with Hardy Potential and Sobolev Critical Exponent Near Zero

Shu Tian

doi:doi:10.11648/j.pamj.20251403.11

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The Asymptotic Behavior of Solutions to Quasilinear Elliptic Equation with Hardy Potential and Sobolev Critical Exponent Near Zero

Shu Tian^*

Published in Pure and Applied Mathematics Journal (Volume 14, Issue 3)

Received: 27 April 2025 Accepted: 28 May 2025 Published: 13 June 2025

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Abstract

This paper mainly explores the precise asymptotic behavior near zero of positive weak solutions to the quasilinear elliptic equation involving Hardy potential and Sobolev critical exponent, which is expressed as under the conditions that , , , , and . The research shows that if is a positive radial weak solution of this equation, then there exists such that , where is the smallest root of the equation . This result accurately depicts the asymptotic characteristics of positive weak solutions of the equation near zero. Compared with previous relevant studies which only indicate that the solutions are bounded near zero, this study further clarifies the limiting situation of the solutions.

Published in	Pure and Applied Mathematics Journal (Volume 14, Issue 3)
DOI	10.11648/j.pamj.20251403.11
Page(s)	29-33
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Quasilinear Elliptic Equations; Hardy Potential: Critical Sobolev Growth

References

[1]	S. Shakerian, J. Vétois, Sharp pointwise estimates for weighted critical p-Laplace equations, Nonlinear Anal. 206 (2021) 112236.
[2]	Y. Li, C. Zhao, A note on exponential decay properties of ground states for quasilinear elliptic equations, Proc. Amer. Math. Soc. 133 (2005) 2005–2012.
[3]	C. He, C. Xiang, Asymptotic behaviors of solutions to quasilinear elliptic equations with Hardy potential, J. Math. Anal. Appl. 441 (2016) 211-234.
[4]	R. Dutta, Apriori Decay Estimates for Hardy-Sobolev-Maz’ya Equations, Calc. Var. Partial Differential Equations 61 (2022) 14.
[5]	P. Ma, S. Huang, Q. Tian, Asymptotic behaviors of solutions to quasilinear elliptic equation with Hardy potential and critical Sobolev exponent, Qual. Theory Dyn. Syst.22 (2023) 153.
[6]	J. Vétois, A priori estimates and application to the symmetry of solutions for critical p-Laplace equations, J. Differential Equations. 260 (2016) 149-161.
[7]	C. Xiang, Gradient estimates for solutions to quasilinear elliptic equations with critical Sobolev growth and Hardy potential, Acta Math. Sci. Ser. B Engl. Ed. 37 (2017) 58-68.
[8]	B.Wang, Z. Zhang, Asymptotic behavior of positive solutions for quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl. 44 (2022) 29.
[9]	F. Esposito, L. Montoro, B. Sciunzi, V. Domenico, Asymptotic behaviour of solutions to the anisotropic doubly critical equation, Calc. Var. Partial Differential Equations 77 (2024) 63.

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APA Style

Tian, S. (2025). The Asymptotic Behavior of Solutions to Quasilinear Elliptic Equation with Hardy Potential and Sobolev Critical Exponent Near Zero. Pure and Applied Mathematics Journal, 14(3), 29-33. https://doi.org/10.11648/j.pamj.20251403.11

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ACS Style

Tian, S. The Asymptotic Behavior of Solutions to Quasilinear Elliptic Equation with Hardy Potential and Sobolev Critical Exponent Near Zero. Pure Appl. Math. J. 2025, 14(3), 29-33. doi: 10.11648/j.pamj.20251403.11

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AMA Style

Tian S. The Asymptotic Behavior of Solutions to Quasilinear Elliptic Equation with Hardy Potential and Sobolev Critical Exponent Near Zero. Pure Appl Math J. 2025;14(3):29-33. doi: 10.11648/j.pamj.20251403.11

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@article{10.11648/j.pamj.20251403.11,
  author = {Shu Tian},
  title = {The Asymptotic Behavior of Solutions to Quasilinear Elliptic Equation with Hardy Potential and Sobolev Critical Exponent Near Zero
},
  journal = {Pure and Applied Mathematics Journal},
  volume = {14},
  number = {3},
  pages = {29-33},
  doi = {10.11648/j.pamj.20251403.11},
  url = {https://doi.org/10.11648/j.pamj.20251403.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20251403.11},
  abstract = {This paper mainly explores the precise asymptotic behavior near zero of positive weak solutions to the quasilinear elliptic equation involving Hardy potential and Sobolev critical exponent, which is expressed as  under the conditions that , , , , and . The research shows that if  is a positive radial weak solution of this equation, then there exists  such that , where  is the smallest root of the equation . This result accurately depicts the asymptotic characteristics of positive weak solutions of the equation near zero. Compared with previous relevant studies which only indicate that the solutions are bounded near zero, this study further clarifies the limiting situation of the solutions.},
 year = {2025}
}

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TY  - JOUR
T1  - The Asymptotic Behavior of Solutions to Quasilinear Elliptic Equation with Hardy Potential and Sobolev Critical Exponent Near Zero

AU  - Shu Tian
Y1  - 2025/06/13
PY  - 2025
N1  - https://doi.org/10.11648/j.pamj.20251403.11
DO  - 10.11648/j.pamj.20251403.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 29
EP  - 33
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20251403.11
AB  - This paper mainly explores the precise asymptotic behavior near zero of positive weak solutions to the quasilinear elliptic equation involving Hardy potential and Sobolev critical exponent, which is expressed as  under the conditions that , , , , and . The research shows that if  is a positive radial weak solution of this equation, then there exists  such that , where  is the smallest root of the equation . This result accurately depicts the asymptotic characteristics of positive weak solutions of the equation near zero. Compared with previous relevant studies which only indicate that the solutions are bounded near zero, this study further clarifies the limiting situation of the solutions.
VL  - 14
IS  - 3
ER  -

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Author Information

Shu Tian

Department of Mathematics, Northwest Normal University, Lanzhou, China

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APA Style

Tian, S. (2025). The Asymptotic Behavior of Solutions to Quasilinear Elliptic Equation with Hardy Potential and Sobolev Critical Exponent Near Zero. Pure and Applied Mathematics Journal, 14(3), 29-33. https://doi.org/10.11648/j.pamj.20251403.11

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ACS Style

Tian, S. The Asymptotic Behavior of Solutions to Quasilinear Elliptic Equation with Hardy Potential and Sobolev Critical Exponent Near Zero. Pure Appl. Math. J. 2025, 14(3), 29-33. doi: 10.11648/j.pamj.20251403.11

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AMA Style

Tian S. The Asymptotic Behavior of Solutions to Quasilinear Elliptic Equation with Hardy Potential and Sobolev Critical Exponent Near Zero. Pure Appl Math J. 2025;14(3):29-33. doi: 10.11648/j.pamj.20251403.11

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@article{10.11648/j.pamj.20251403.11,
  author = {Shu Tian},
  title = {The Asymptotic Behavior of Solutions to Quasilinear Elliptic Equation with Hardy Potential and Sobolev Critical Exponent Near Zero
},
  journal = {Pure and Applied Mathematics Journal},
  volume = {14},
  number = {3},
  pages = {29-33},
  doi = {10.11648/j.pamj.20251403.11},
  url = {https://doi.org/10.11648/j.pamj.20251403.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20251403.11},
  abstract = {This paper mainly explores the precise asymptotic behavior near zero of positive weak solutions to the quasilinear elliptic equation involving Hardy potential and Sobolev critical exponent, which is expressed as  under the conditions that , , , , and . The research shows that if  is a positive radial weak solution of this equation, then there exists  such that , where  is the smallest root of the equation . This result accurately depicts the asymptotic characteristics of positive weak solutions of the equation near zero. Compared with previous relevant studies which only indicate that the solutions are bounded near zero, this study further clarifies the limiting situation of the solutions.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - The Asymptotic Behavior of Solutions to Quasilinear Elliptic Equation with Hardy Potential and Sobolev Critical Exponent Near Zero

AU  - Shu Tian
Y1  - 2025/06/13
PY  - 2025
N1  - https://doi.org/10.11648/j.pamj.20251403.11
DO  - 10.11648/j.pamj.20251403.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 29
EP  - 33
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20251403.11
AB  - This paper mainly explores the precise asymptotic behavior near zero of positive weak solutions to the quasilinear elliptic equation involving Hardy potential and Sobolev critical exponent, which is expressed as  under the conditions that , , , , and . The research shows that if  is a positive radial weak solution of this equation, then there exists  such that , where  is the smallest root of the equation . This result accurately depicts the asymptotic characteristics of positive weak solutions of the equation near zero. Compared with previous relevant studies which only indicate that the solutions are bounded near zero, this study further clarifies the limiting situation of the solutions.
VL  - 14
IS  - 3
ER  -

Copy | Download