On the Poisson equation of p-Laplacian and the nonlinear Hardy-type problems


In this note, we show that in some cases, via the use of Hardy-type inequality, there is a non-trivial nonnegative \(W^{1,p}(R^n)\) weak solution to quasi-linear elliptic problem with the p-Laplacian on \(R^n\) and with Hardy-type singularity term. We also study the behavior of solutions to the Poisson equation of p-Laplacian on the whole space and this Poisson equation has a close relationship with the Gelfand-type equation.

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  1. 1.

    Bartsch, T., Wang, Z.Q.: Existence and multiplicity results for some superlinear elliptic problems on \(R^3\). Commun. Partial Differ. Equ. 20(9–10), 1725–1741 (1995)

    Article  Google Scholar 

  2. 2.

    Caffarelli, L., Garofalo, N., Segala, F.: A gradient bound for entire solutions of quasi-linear equations and its consequences. Commun. Pure Appl. Math. 47(11), 1457–1473 (1994). (Reviewer: C. A. Swanson) 35B45 (35J60 53A10)

  3. 3.

    Chen, W., Li, C.: Methods on Nonlinear Elliptic Equations. AIMS Book Series, vol. 4 (2010)

  4. 4.

    Costa, D.G.: On a class of elliptic systems in \(R^N\). Electron. J. Differ. Equ. 7, 1–14 (1994)

    Google Scholar 

  5. 5.

    Damascelli, L., Pardo, R.: A priori estimates for some elliptic equations involving the \(p\)-Laplacian. Nonlinear Anal. Real World Appl. 41, 475–496 (2018). 35J92 (35B33 35B45)

  6. 6.

    Damascelli, L., Merchan, S., Montoro, L., Sciunzi, B.: Radial symmetry and applications for a problem involving the \({\Delta }_{p}\) operator and critical nonlinearity in \(R^N\). Adv. Math. 265, 313–335 (2014). 35J92 (35B06 35B09 35B33)

  7. 7.

    Damascelli, L., Pacella, F.: Monotonicity and symmetry results for p-Laplace equations and applications. Adv. Differ. Equ. 5(7–9), 1179–1200. (Reviewer: Srinivasan Kesavan) 35J65 (35B05 35B50 35J70) (2000)

  8. 8.

    Frank, R.L., Seiringer, R.: Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255, 3407–3430 (2008)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Guo, Z.M., Ma, L.: Asymptotic behavior of positive solutions of some quasilinear elliptic problems. J. Lond. Math. Soc. (2) 76(2), 419–437 (2007)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Kazdan, J., Warner, F.: Curvature functions for compact two manifolds. Ann. Math. 99, 14–47 (1974)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics, vol. 14, 2nd edn. American Mathematical Society, Providence (2001)

    Google Scholar 

  12. 12.

    Ma, L.: Boundedness of solutions to Ginzburg–Landau fractional Laplacian equation. Int. J. Math. 27(5), 1650048 (2016)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Ma, L.: On nonlocal nonlinear elliptic problems with the fractional Laplacian. Glasgow Math. J. 62, 75–84 (2020). https://doi.org/10.1017/S0017089518000538

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Ma, L., Chen, D.: Radial symmetry and monotonicity results for an integral equation. J. Math. Anal. Appl. 342, 943–949 (2008)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Ma, L., Zhao, L.: Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Rational Mech. Anal. 195, 455–467 (2010)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Ma, L., Ning, S.: Existence, multiplicity, and stability results for positive solutions of non-linear p-Laplacian equations. Chin. Ann. Math. 25B, 275–286 (2004)

    Article  Google Scholar 

  17. 17.

    Ma, L., Zhang, K.S.: ground states of a nonlinear drifting Schodinger equation. Appl. Math. Lett. 105, 106324 (2020)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Peral, I.: Some results on quasilinear elliptic equations: growth versus shape. Nonlinear functional analysis and applications to differential equations (Trieste, 1997), 153–202, World Sci. Publ., River Edge, NJ (1998). (Reviewer: Mei Jun Zhu) 35J65 (58E07)

  19. 19.

    Talenti, G.: Rearrangements of functionals and partial differential equations. In: Fasano, A., Primicerio, M. (eds.) Nonlinear Diffusion Problems. Springer Lecture Notes in Mathematics, p. 1224 (1985)

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Correspondence to Li Ma.

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The research is partially supported by the National Natural Science Foundation of China (No. 11771124). We certify that the general content of the manuscript, in whole or in part, is not submitted, accepted, or published elsewhere, including conference proceedings.

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Ma, L. On the Poisson equation of p-Laplacian and the nonlinear Hardy-type problems. Z. Angew. Math. Phys. 72, 34 (2021). https://doi.org/10.1007/s00033-020-01465-8

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  • p-Laplacian
  • Ground state
  • Existence of positive solutions
  • Rearrangement
  • Variational method
  • Nehari manifold

Mathematics Subject Classification

  • 35A05
  • 35A15
  • 35B50
  • 35J60
  • 46Txx
  • 53C70
  • 58E50