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Articles

Nonexistence of positive solutions to a system of elliptic inequalities involving the Grushin operator

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Pages 372-384 | Received 18 May 2021, Accepted 15 Oct 2021, Published online: 11 Nov 2021

References

  • Grushin VV. On a class of elliptic pseudo differential operators degenerate on a submanifold. Math USSR-Sborn. 1971;13(2):155.
  • Baouendi MS. Sur une classe d'opérateurs elliptiques dégénérés. Bull Soc Math France. 1967;95:45–87.
  • Franchi B, Gutiérrez CE, Wheeden RL. Weighted Sobolev-Poincaré inequalities for Grushin type operators. Commun Partial Differ Equ. 1994;19(3-4):523–604.
  • Jerison D, Lee JM. The Yamabe problem on CR manifolds. J Differ Geom. 1987;25(2):167–197.
  • Birindelli I, Capuzzo Dolcetta I, Cutrı A. Liouville theorems for semilinear equations on the Heisenberg group. Ann Inst H Poincaré Anal Non Linéaire. 1997;14(3):295–308.
  • Birindelli I, Prajapat J. Nonlinear Liouville theorems in the Heisenberg group via the moving plane method. Commun Partial Differ Equ. 1999;24(9–10):1875–1890.
  • Garofalo N, Vassilev D. Symmetry properties of positive entire solutions of Yamabe-type equations on groups of Heisenberg type. Duke Math J. 2001;106(3):411–448.
  • Monticelli DD. Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators. J Eur Math Soc. 2010;12(3):611–654.
  • Armstrong SN, Sirakov B. Nonexistence of positive supersolutions of elliptic equations via the maximum principle. Commun Partial Differ Equ. 2011;36(11):2011–2047.
  • Mitidieri E, Pohozaev SI. A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities. Tr Mat Inst Stekl. 2001;234:1–384.
  • D'Ambrosio L, Mitidieri E. A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities. Adv Math. 2010;224(3):967–1020.
  • D'Ambrosio L, Lucente S. Nonlinear Liouville theorems for Grushin and Tricomi operators. J Differ Equ. 2003;193(2):511–541.
  • Capuzzo Dolcetta I, Cutri A. On the Liouville property for sublaplacians. Ann Scuol Normal Super Pisa Class Sci. 1997;25(1–2):239–256.
  • Le P, Duong AT, Nguyen NT. Liouville-type theorems for sub-elliptic systems involving Δλ-laplacian. Complex Var Ellip Equ. DOI:10.1080/17476933.2020.1816981
  • Franchi B, Lanconelli EUne métrique associée à une classe d'opérateurs elliptiques dégénérés. Rend. Sem. Mat. Univ. Politec. Torino, Special Issue (1983). p. 105–114 (1984). Conference on linear partial and pseudodifferential operators (Torino, 1982).
  • Kogoj AE, Lanconelli E. On semilinear Δλ-Laplace equation. Nonlinear Anal. 2012;75(12):4637–4649.
  • Kogoj AE, Sonner S. Hardy type inequalities for Δλ-Laplacians. Complex Var Ellip Equ. 2016;61(3):422–442.
  • Luyen DT, Tri NM. Existence of solutions to boundary-value problems for similinear Δγ differential equations. Math Notes. 2015;97(1-2):73–84.
  • Rahal B. Liouville-type theorems with finite Morse index for semilinear Δλ-Laplace operators. Nonlinear Differ Equ Appl. 2018;25. Paper No. 21.
  • Kogoj AE, Lanconelli E. Linear and semilinear problems involving Δλ-Laplacians. In: Proceedings of the International Conference ‘Two nonlinear days in Urbino 2017’. Electron. J. Differ. Equ. Conf., Vol. 25. San Marcos, TX: Texas State Univ.–San Marcos, Dept. Math.; 2018. p. 167–178.
  • Wang C, Wang Q, Yang J. On the Grushin critical problem with a cylindrical symmetry. Adv. Differ Equ. 2015;20(1–2):77–116.
  • Monti R, Morbidelli D. Kelvin transform for Grushin operators and critical semilinear equations. Duke Math J. 2006;131(1):167–202.
  • Garofalo N, Lanconelli E. Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation. Ann Inst Fourier. 1990;40(2):313–356.
  • Anh CT, My BK. Liouville-type theorems for elliptic inequalities involving the Δλ-Laplace operator. Complex Var Ellip Equ. 2016;61(7):1002–1013.
  • Serrin J, Zou H. Non-existence of positive solutions of Lane-Emden systems. Differ Integr Equ. 1996;9(4):635–653.
  • Yu X. Liouville type theorem for nonlinear elliptic equation involving Grushin operators. Commun Contemp Math. 2015;17(5):1450050, 12.
  • Morbidelli D. Liouville theorem, conformally invariant cones and umbilical surfaces for Grushin-type metrics. Israel J Math. 2009;173:379–402.
  • Gidas B, Spruck J. Global and local behavior of positive solutions of nonlinear elliptic equations. Commun Pure Appl Math. 1981;34(4):525–598.
  • Chen W, Li C. Classification of solutions of some nonlinear elliptic equations. Duke Math J. 1991;63(3):615–622.
  • Duong AT, Nguyen NT. Liouville type theorems for elliptic equations involving Grushin operator and advection. Electron J Differ Equ. 2017;Paper No. 108.
  • Le P. Liouville theorems for stable weak solutions of elliptic problems involving Grushin operator. Commun Pure Appl Anal. 2020;19(1):511–525.
  • Wei Y, Chen C, Chen Q, et al. Liouville-type theorem for nonlinear elliptic equations involving p-Laplace-type Grushin operators. Math Methods Appl Sci. 2020;43(1):320–333.
  • D'Ambrosio L. Hardy inequalities related to Grushin type operators. Proc Am Math Soc. 2004;132(3):725–734.
  • Yang Q, Su D, Kong Y. Improved Hardy inequalities for Grushin operators. J Math Anal Appl. 2015;424(1):321–343.
  • Bonfiglioli A, Lanconelli E, Tommasoli A. Convexity of average operators for subsolutions to subelliptic equations. Anal PDE. 2014;7(2):345–373.
  • Monti R. Sobolev inequalities for weighted gradients. Commun Partial Differ Equ. 2006;31(10–12):1479–1504.
  • Franchi B, Lanconelli E. An embedding theorem for Sobolev spaces related to nonsmooth vector fields and Harnack inequality. Commun Partial Differ Equ. 1984;9(13):1237–1264.

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