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On the p-Laplacian Lichnerowicz equation on compact Riemannian manifolds

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Abstract

In this paper, we deal with a singular quasilinear critical elliptic equation of Lichnerowicz type involving the p-Laplacian operator. With the help of the subcritical approach from the variational method, we obtain the non-existence, existence, and multiplicity results under some given assumptions.

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References

  1. Aubin T. Problèmes isopérimétriques et espaces de Sobolev. J Differential Geom, 1976, 11: 573–598

    Article  Google Scholar 

  2. Aubin T. Some Nonlinear Problems in Riemannian Geometry. Springer Monographs in Mathematics. Berlin: Springer-Verlag, 1998

    Book  Google Scholar 

  3. Benalili M. Nodal solutions to quasilinear elliptic equations on compact Riemannian manifolds. Commun Contemp Math, 2010, 12: 909–937

    Article  MathSciNet  Google Scholar 

  4. Benalili M, Maliki Y. Solving p-Laplacian equations on complete manifolds. Electron J Differential Equations, 2006, 2006: 1–9

    MathSciNet  MATH  Google Scholar 

  5. Benalili M, Maliki Y. Multiplicity of solutions for elliptic quasilinear equations with critical exponent on compact manifolds. Nonlinear Anal, 2009, 71: 5946–5960

    Article  MathSciNet  Google Scholar 

  6. Chen N B, Liu X C. A quasilinear elliptic equation with critical growth on compact Riemannian manifold. J Pseudo-Differ Oper Appl, 2019, 10: 975–997

    Article  MathSciNet  Google Scholar 

  7. Choquet-Bruhat Y, Isenberg J, Pollack D. The Einstein-scalar field constraints on asymptotically Euclidean manifolds. Chin Ann Math Ser B, 2006, 27: 31–52

    Article  MathSciNet  Google Scholar 

  8. Choquet-Bruhat Y, Isenberg J, Pollack D. The constraint equations for the Einstein-scalar field system on compact manifolds. Classical Quantum Gravity, 2007, 24: 809–828

    Article  MathSciNet  Google Scholar 

  9. Demengel F, Hebey E. On some nonlinear equations involving the p-Laplacian with critical Sobolev growth. Adv Differential Equations, 1998, 3: 533–574

    MathSciNet  MATH  Google Scholar 

  10. Druet O. Generalized scalar curvature type equations on compact Riemannian manifolds. Proc Roy Soc Edinburgh Sect A, 2000, 130: 767–788

    Article  MathSciNet  Google Scholar 

  11. Druet O, Hebey E. Stability and instability for Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds. Math Z, 2009, 263: 33–67

    Article  MathSciNet  Google Scholar 

  12. Guedda M, Veron L. Quasilinear elliptic equations involving critical Sobolev exponents. Nonlinear Anal, 1989, 13: 879–902

    Article  MathSciNet  Google Scholar 

  13. Hebey E. Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities. Courant Lecture Notes, vol. 5. New York: Courant Inst Math Sci, 1999

    MATH  Google Scholar 

  14. Hebey E, Pacard F, Pollack D. A variational analysis of Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds. Comm Math Phys, 2008, 278: 117–132

    Article  MathSciNet  Google Scholar 

  15. Isenberg J. Constant mean curvature solutions of the Einstein constraint equations on closed manifolds. Classical Quantum Gravity, 1995, 12: 2249–2274

    Article  MathSciNet  Google Scholar 

  16. Ladyzhenskaya O A, Uraltseva N N. Linear and Quasilinear Elliptic Equations. New York-London: Academic Press, 1968

    Google Scholar 

  17. Lindqvist P. On the equation div(|∇u|p−2∇u) + λ|u|p−2u = 0. Proc Amer Math Soc, 1990, 109: 157–164

    MathSciNet  MATH  Google Scholar 

  18. Ma L. Liouville type theorem and uniform bound for the Lichnerowicz equation and the Ginzburg-Landau equation. C R Math Acad Sci Paris, 2010, 348: 993–996

    Article  MathSciNet  Google Scholar 

  19. Ma L, Sun Y H, Tang Y. Heat flow method for Lichnerowicz type equations on closed manifolds. Z Angew Math Phys, 2012, 63: 261–270

    Article  MathSciNet  Google Scholar 

  20. Ma L, Wei J C. Stability and multiple solutions to Einstein-scalar field Lichnerowicz equation on manifolds. J Math Pures Appl (9), 2013, 99: 174–186

    Article  MathSciNet  Google Scholar 

  21. Ma L, Xu X W. Uniform bound and a non-existence result for Lichnerowicz equation in the whole n-space. C R Math Acad Sci Paris, 2009, 347: 805–808

    Article  MathSciNet  Google Scholar 

  22. Maliki Y. Existence and multiplicity results for nonlinear critical Neumann problem on compact Riemannian manifolds. NoDEA Nonlinear Differential Equations Appl, 2013, 20: 1–22

    Article  MathSciNet  Google Scholar 

  23. Ngô Q A. Einstein constraint equations on Riemannian manifolds. In: Geometric Analysis around Scalar Curvatures, vol. 31. Singapore: World Scientific, 2016, 119–210

    Chapter  Google Scholar 

  24. Ngô Q A, Xu X W. Existence results for the Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds. Adv Math, 2012, 230: 2378–2415

    Article  MathSciNet  Google Scholar 

  25. Ngô Q A, Xu X W. Existence results for the Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds in the positive case. Bull Inst Math Acad Sin (NS), 2014, 9: 451–485

    MathSciNet  MATH  Google Scholar 

  26. Ngô Q A, Xu X W. Existence results for the Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds in the null case. Comm Math Phys, 2015, 334: 193–222

    Article  MathSciNet  Google Scholar 

  27. Premoselli B. Effective multiplicity for the Einstein-scalar field Lichnerowicz equation. Calc Var Partial Differential Equations, 2015, 53: 29–64

    Article  MathSciNet  Google Scholar 

  28. Rauzy A. Courbures scalaires des varietes d’invariant conforme negatif. Trans Amer Math Soc, 1995, 347: 4729–4745

    MathSciNet  MATH  Google Scholar 

  29. Silva C, Pina R, Souza M. On the study of a class of non-linear differential equations on compact Riemannian manifolds. Publ Math Debrecen, 2018, 92: 277–292

    Article  MathSciNet  Google Scholar 

  30. Song X F, Zhao L. Gradient estimates for the elliptic and parabolic Lichnerowicz equations on compact manifolds. Z Angew Math Phys, 2010, 61: 655–662

    Article  MathSciNet  Google Scholar 

  31. Struwe M. Variational Methods, 2nd ed. Berlin: Springer, 1999

    MATH  Google Scholar 

  32. Talenti G. Best constant in Sobolev inequality. Ann Mat Pura Appl (4), 1976, 110: 353–372

    Article  MathSciNet  Google Scholar 

  33. Tolksdorf P. Regularity for a more general class of quasilinear elliptic equations. J Differential Equations, 1984, 51: 126–150

    Article  MathSciNet  Google Scholar 

  34. Zhao L. Liouville theorem for Lichnerowicz equation on complete noncompact manifolds. Funkcial Ekvac, 2014, 57: 163–172

    Article  MathSciNet  Google Scholar 

  35. Zhao L, Wang L. Liouville theorem for p-Laplacian Lichnerowicz equation on compact manifolds. J Geom Phys, 2017, 121: 8–14

    Article  MathSciNet  Google Scholar 

  36. Zhao L, Yang D. Gradient estimates for the p-Laplacian Lichnerowicz equation on smooth metric measure spaces. Proc Amer Math Soc, 2018, 146: 5451–5461

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11771342 and 11571259), and the Natural Science Foundation of Hubei Province (Grant No. 2019CFA007).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Nanbo Chen & Xiaochun Liu

  2. Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan, 430072, China

    Xiaochun Liu

Authors
  1. Nanbo Chen
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  2. Xiaochun Liu
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Correspondence to Xiaochun Liu.

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Chen, N., Liu, X. On the p-Laplacian Lichnerowicz equation on compact Riemannian manifolds. Sci. China Math. 64, 2249–2274 (2021). https://doi.org/10.1007/s11425-020-1679-5

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  • DOI: https://doi.org/10.1007/s11425-020-1679-5

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