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Asymptotic Behaviors of Solutions to Quasilinear Elliptic Equation with Hardy Potential and Critical Sobolev Exponent

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Abstract

In this paper, we mainly consider the asymptotic behaviors of positive weak solutions to the following critical quasilinear elliptic equation with Hardy potential

$$\begin{aligned} -\text {div}\left( \frac{|\nabla u|^{p-2}\nabla u}{|x|^{ap}}\right) -\gamma \frac{ u^{p-1}}{|x|^{(a+1)p}}=\frac{u^{p^{*}_{a,b}-1}}{|x|^{bp^{*}_{a,b}}}, ~~~x\in \mathbb {R}^{N}, \end{aligned}$$

where \(1<p<N\), \(0\leqslant a<\frac{N-p}{p}\), \(a\leqslant b<a+1\), \(0\leqslant \gamma <\left( \frac{N-(a+1)p}{p}\right) ^{p}\) and \(p^{*}_{a,b}=\frac{Np}{N-(a+1-b)p}\). The main results of this paper generalize the some related works of [8, 23].

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References

  1. Brasco, L., Mosconi, S., Squassina, M.: Optimal decay of extremals for the fractional Sobolev inequality. Calc. Var. Partial Diff. Equ. 55, 23 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brezis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical exponents. Commun. Pure Appl. Math. 36, 437–477 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cao, D., Han, P.: Solutions to critical elliptic equations with multi-singular inverse square potentials. J. Differ. Equ. 224, 332–372 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cao, D., Yan, S.: Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential. Calc. Var. Partial Differ. Equ. 38, 471–501 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dutta, R.: Apriori decay estimates for Hardy-Sobolev-Mazya equations. Calc. Var. Partial Differ. Equ. 61, 1–36 (2022)

    Article  Google Scholar 

  6. Feng, Z., Tan, C., Wei, L.: Uniqueness and asymptotic behavior of positive solution of quasilinear elliptic equations with Hardy potential. Nonlinear Anal. 202, 112152 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  7. Han, P.: Asymptotic behavior of solutions to semilinear elliptic equations with Hardy potential. Proc. Am. Math. Soc. 135, 365–372 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Han, Q., Lin, F.: Elliptic Partial Differential Equations, p. 123. Courant Institute of Mathematical Sciences, American Mathematical Society (1997)

    MATH  Google Scholar 

  9. He, C., Xiang, C.: Asymptotic behaviors of solutions to quasilinear elliptic equations with Hardy potential. J. Math. Anal. Appl. 441, 211–234 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Itakura, K., Tanaka, W.: A note on the asymptotic behavior of radial solutions to quasilinear elliptic equations with a Hardy potential. Proc. Am. Math. Soc. 8, 302–310 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kang, D.: On the quasilinear elliptic problems with critical Sobolev-Hardy exponents and Hardy terms. Nonlinear Anal. 68, 1973–1985 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lindqvist, P.: Notes on the \(p\)-Laplace equation, No. 161, University of Jyväskylä (2017)

  13. Loiudice, A.: Critical problems with hardy potential on Stratified Lie groups. Adv. Differ. Equ. 28, 1–33 (2023)

    MathSciNet  MATH  Google Scholar 

  14. Ou, Q.: On the classification of entire solutions to the critical \(p\)-Laplace equation arXiv:2210.05141 (2022)

  15. Roncoroni, A.: An overview on extremals and critical points of the Sobolev inequality in convex cones. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 33, 967–995 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  16. Shen, Y.: Existence of solutions to elliptic problems with fractional \(p\)-Laplacian and multiple critical nonlinearities in the entire space \(\mathbb{R} ^{N}\). Nonlinear Anal. 202, 112102 (2021)

    Article  MATH  Google Scholar 

  17. Shakerian, S., Vétois, J.: Sharp pointwise estimates for weighted critical \(p\)-Laplace equations. Nonlinear Anal. 206, 112236 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  18. Vétois, J.: A priori estimates and application to the symmetry of solutions for critical \(p\)-Laplace equations. J. Differ. Equ. 260, 149–161 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wei, L., Cheng, X., Feng, Z.: Exact behavior of positive solutions to elliptic equations with multi-singular inverse square potentials. Discr. Contin. Dyn. Syst. 36, 7169–7189 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wei, L., Du, Y.: Exact singular behavior of positive solutions to nonlinear elliptic equations with a Hardy potential. J. Differ. Equ. 262, 3864–3886 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wei, L., Feng, Z.: Isolated singularity for semilinear elliptic equations. Discr. Contin. Dyn. Syst. 35, 3239–3252 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wang, B., Zhang, Z.: Asymptotic behavior of positive solutions for quasilinear elliptic equations. Nonlinear Differ. Equ. Appl. 4, 29 (2022)

    Article  Google Scholar 

  23. Xiang, C.: Asymptotic behaviors of solutions to quasilinear elliptic equations with critical Sobolev growth and Hardy potential. J. Differ. Equ. 259, 3929–3954 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Xuan, B., Wang, J.: Asymptotic behavior of extremal functions to an inequality involving Hardy potential and critical Sobolev exponent arXiv:math/0701687

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Acknowledgements

This works was partially supported by the National Natural Science Foundation of China (No. 11761059), Fundamental Research Funds for the Central Universities (No. 31920220067).

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Authors and Affiliations

  1. School of Mathematics and Computer Science, Northwest Minzu University, Lanzhou, 730030, Gansu, People’s Republic of China

    Ma Pu, Shuibo Huang & Qiaoyu Tian

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  1. Ma Pu
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  2. Shuibo Huang
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M. Pu and S. Huang wrote the main manuscript text, and Q. Tian modified the article. All authors reviewed the manuscript.

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Correspondence to Shuibo Huang.

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Pu, M., Huang, S. & Tian, Q. Asymptotic Behaviors of Solutions to Quasilinear Elliptic Equation with Hardy Potential and Critical Sobolev Exponent. Qual. Theory Dyn. Syst. 22, 153 (2023). https://doi.org/10.1007/s12346-023-00847-3

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