Home Nonnegative solutions of a fractional differential inequality on Grushin spaces and nilpotent Lie groups
Article
Licensed
Unlicensed Requires Authentication

Nonnegative solutions of a fractional differential inequality on Grushin spaces and nilpotent Lie groups

  • Nan Zhao and Yu Liu EMAIL logo
Published/Copyright: August 30, 2022

Abstract

In this paper, we investigate the nonnegative solutions of the differential inequality

u p s u

on the Grushin space 𝔾 α n for ( p , s , α ) ( 1 , ) × ( 0 , 1 ) × ( 0 , ) , where the s are the fractional powers of the Grushin operator . We show that any nonnegative solution of the fractional order differential inequality displayed above is zero if and only if p Q Q - 2 s , where Q is the homogeneous dimension of 𝔾 α n . Moreover, we also consider the similar problems of nonnegative weak solutions of the fractional sub-Laplacian differential inequality on nilpotent Lie groups.


Communicated by Christopher D. Sogge


Award Identifier / Grant number: 11671031

Award Identifier / Grant number: Z17111000220000

Funding statement: The second author was supported by the National Natural Science Foundation of China (No. 11671031) and Beijing Municipal Science and Technology Project (No. Z17111000220000).

References

[1] A. V. Balakrishnan, Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math. 10 (1960), 419–437. 10.2140/pjm.1960.10.419Search in Google Scholar

[2] R. Balhara, P. Boggarapu and S. Thangavelu, An extension problem and Hardy type inequalities for the Grushin operator, Geometric Aspects of Harmonic Analysis, Springer INdAM Ser. 45, Springer, Cham (2021), 1–28. 10.1007/978-3-030-72058-2_1Search in Google Scholar

[3] I. Birindelli, I. Capuzzo Dolcetta and A. Cutrì, Liouville theorems for semilinear equations on the Heisenberg group, Ann. Inst. H. Poincaré C Anal. Non Linéaire 14 (1997), no. 3, 295–308. 10.1016/s0294-1449(97)80138-2Search in Google Scholar

[4] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 7–9, 1245–1260. 10.1080/03605300600987306Search in Google Scholar

[5] D. Chamorro and O. Jarrín, Fractional Laplacians, extension problems and Lie groups, C. R. Math. Acad. Sci. Paris 353 (2015), no. 6, 517–522. 10.1016/j.crma.2015.04.007Search in Google Scholar

[6] K.-J. Engel and R. Nagel, A Short Course on Operator Semigroups, Universitext, Springer, New York, 2006. Search in Google Scholar

[7] F. Ferrari and B. Franchi, Harnack inequality for fractional sub-Laplacians in Carnot groups, Math. Z. 279 (2015), no. 1–2, 435–458. 10.1007/s00209-014-1376-5Search in Google Scholar

[8] B. Franchi and E. Lanconelli, Une métrique associée à une classe d’opérateurs elliptiques dégénérés, Conference on Linear Partial and Pseudodifferential Operators (Torino 1982), Università e Politecnico, Torino (1983), 105–114, Search in Google Scholar

[9] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), no. 4, 525–598. 10.1002/cpa.3160340406Search in Google Scholar

[10] A. Grigor’yan and Y. Sun, On nonnegative solutions of the inequality Δ u + u σ 0 on Riemannian manifolds, Comm. Pure Appl. Math. 67 (2014), no. 8, 1336–1352. 10.1002/cpa.21493Search in Google Scholar

[11] A. Grigor’yan and Y. Sun, On positive solutions of semi-linear elliptic inequalities on Riemannian manifolds, Calc. Var. Partial Differential Equations 58 (2019), no. 6, Paper No. 207. 10.1007/s00526-019-1645-6Search in Google Scholar

[12] M. Gromov, Carnot–Carathéodory spaces seen from within, Sub-Riemannian Geometry, Progr. Math. 144, Birkhäuser, Basel (1996), 79–323. 10.1007/978-3-0348-9210-0_2Search in Google Scholar

[13] N. Jacob, Pseudo Differential Operators and Markov Processes. Vol. I, Imperial College, London, 2001. 10.1142/p245Search in Google Scholar

[14] G. Karisti, E. Mitidieri and S. I. Pokhozhaev, Liouville theorems for quasilinear elliptic inequalities, Dokl. Math. 79 (2009), no. 6, 118–124. 10.1134/S1064562409010360Search in Google Scholar

[15] Y. Liu, Y. Wang and J. Xiao, Nonnegative solutions of a fractional sub-Laplacian differential inequality on Heisenberg group, Dyn. Partial Differ. Equ. 12 (2015), no. 4, 379–403. 10.4310/DPDE.2015.v12.n4.a4Search in Google Scholar

[16] L. Lorenzi and M. Bertoldi, Analytical Methods for Markov Semigroups, Pure Appl. Math. (Boca Raton) 283, Chapman & Hall/CRC, Boca Raton, 2006. 10.1201/9781420011586Search in Google Scholar

[17] R. Monti and D. Morbidelli, Isoperimetric inequality in the Grushin plane, J. Geom. Anal. 14 (2004), no. 2, 355–368. 10.1007/BF02922077Search in Google Scholar

[18] R. Monti and F. Serra Cassano, Surface measures in Carnot–Carathéodory spaces, Calc. Var. Partial Differential Equations 13 (2001), no. 3, 339–376. 10.1007/s005260000076Search in Google Scholar

[19] A. Nagel, E. M. Stein and S. Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), no. 1–2, 103–147. 10.1007/BF02392539Search in Google Scholar

[20] D. W. Robinson and A. Sikora, Analysis of degenerate elliptic operators of Grušin type, Math. Z. 260 (2008), no. 3, 475–508. 10.1007/s00209-007-0284-3Search in Google Scholar

[21] D. W. Robinson and A. Sikora, The limitations of the Poincaré inequality for Grušin type operators, J. Evol. Equ. 14 (2014), no. 3, 535–563. 10.1007/s00028-014-0227-5Search in Google Scholar

[22] J. Serrin and H. Zou, Cauchy–Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math. 189 (2002), no. 1, 79–142. 10.1007/BF02392645Search in Google Scholar

[23] P. R. Stinga and J. L. Torrea, Extension problem and Harnack’s inequality for some fractional operators, Comm. Partial Differential Equations 35 (2010), no. 11, 2092–2122. 10.1080/03605301003735680Search in Google Scholar

[24] Y. Sun, Uniqueness result for non-negative solutions of semi-linear inequalities on Riemannian manifolds, J. Math. Anal. Appl. 419 (2014), no. 1, 643–661. 10.1016/j.jmaa.2014.05.011Search in Google Scholar

[25] Y. Sun, On nonexistence of positive solutions of quasi-linear inequality on Riemannian manifolds, Proc. Amer. Math. Soc. 143 (2015), no. 7, 2969–2984. 10.1090/S0002-9939-2015-12705-0Search in Google Scholar

[26] Y. Sun, On the uniqueness of nonnegative solutions of differential inequalities with gradient terms on Riemannian manifolds, Commun. Pure Appl. Anal. 14 (2015), no. 5, 1743–1757. 10.3934/cpaa.2015.14.1743Search in Google Scholar

[27] N. T. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, Cambridge Tracts in Math. 100, Cambridge University, Cambridge, 1992. 10.1017/CBO9780511662485Search in Google Scholar

[28] Y. Wang and J. Xiao, A constructive approach to positive solutions of Δ p u + f ( u , u ) 0 on Riemannian manifolds, Ann. Inst. H. Poincaré C Anal. Non Linéaire 33 (2016), no. 6, 1497–1507. 10.1016/j.anihpc.2015.06.003Search in Google Scholar

[29] Y. Wang and J. Xiao, A uniqueness principle for u p ( - Δ ) α 2 u in the Euclidean space, Commun. Contemp. Math. 18 (2016), no. 6, Article ID 1650019. 10.1142/S021919971650019XSearch in Google Scholar

[30] J.-M. Wu, Geometry of Grushin spaces, Illinois J. Math. 59 (2015), no. 1, 21–41. 10.1215/ijm/1455203157Search in Google Scholar

Received: 2022-04-16
Revised: 2022-07-04
Published Online: 2022-08-30
Published in Print: 2023-01-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 16.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2022-0120/html
Scroll to top button