Generalised Picone's identity and some Qualitative properties of p-sub-Laplacian on Heisenberg group

Abstract

In this article, we derive a generalised nonlinear Picone's identity for $p$ sub-Laplacian on the Heisenberg group. Our main result generalises the Picone's identity established by Niu et al.(Proceedings of the American Mathematical Society , Dec., 2001, Vol. 129, No. 12, pp. 3623-3630). As an application of Picone's identity, we prove a Hardy type inequality and Picone's inequality. We also establish some qualitative results involving the system of nonlinear equations involving $p$-sub-Laplacian.

Keywords

• Picone' Identity
• Heisenberg Group
• Hardy type inequality
• Picone's Inequality

Kaynakça

1. [1] W. Allegretto, Positive solutions and spectral properties of weakly coupled elliptic systems, J. Math. Anal. Appl., 120(2) (1986), 723-729.
2. [2] W. Allegretto, On the principal eigenvalues of indenite elliptic problems, Math. Z. 195(1) (1987), 29-35.
3. [3] W. Allegretto, Sturmian theorems for second order systems, Proc. Amer. Math. Soc. 94(2) (1985), 291-296.
4. [4] W. Allegretto and Y.X.Huang, A Picone's identity for the p-Laplacian and applications, Nonlinear Anal., 32(7) (1998), 819-830.
5. [5] K.Bal, Generalized Picone's identity and its applications, Electron. J. Diff. Equations., no. 243 (2013), 1-6.
6. [6] G. Bognár, O. Doslý, Picone-type identity for pseudo p-Laplacian with variable power, Electron. J. Diff. Equations 2012, No. 174, 1-8.
7. [7] A. Bonfiglioli, E. Lanconelli, F. Uguzzoni, Stratified Lie groups and potential theory for their sub-Laplacians, Springer Science & Business Media, 2007.
8. [8] J.M. Bony, Principe du maximum, inégalité de Harnack et unicité du probleme de Cauchy pour les opérateurs elliptiques dégénérés, Annales de l'institut Fourier, 19(1), 1969, 277-304.

9. [9] L. Capogna, D. Danielli, S. D. Pauls, J. Tyson, An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem, Vol. 259, Springer Science & Business Media, 2007.
10. [10] J. Dou, Picone inequalities for p-sublaplacian on the Heisenberg group and its applications, Commun. Contem. Math. 12 No. 2 (2010), 295-307.
11. [11] G. Dwivedi, J. Tyagi, Remarks on the qualitative questions for biharmonic operators, Taiwan. J. Math., 19(6), (2015), 1743-1758.
12. [12] G.Dwivedi, J.Tyagi, A note on the Caccioppoli inequality for biharmonic operators, Mediterr. J. Math. 13,(4), (2016), 1823-1828.
13. [13] G. Dwivedi, J. Tyagi, Picone's identity for biharmonic operators on Heisenberg group and its applications, Nonlinear Differential Equations Appl. NoDEA, 23(2), (2016).
14. [14] T. Feng, A new nonlinear Picone identity and applications, Math. Appl., 30(2), (2017), 278-283.
15. [15] T. Feng, A generalized nonlinear Picone identity for the p-biharmonic operator and its applications, J. Inequal. Appl., 2019, (2019):56.
16. [16] Y. Han, P. Niu, Some Hardy type inequalities in the Heisenberg group, JIPAM. J. Inequal. Pure Appl. Math 4, (2003), 1-5.
17. [17] J. Han, P. Niu, W. Qin, Hardy inequalities in half spaces of the Heisenberg group, Bull. Korean Math. Soc., 45(3), (2008), 405-417.
18. [18] J. Jaros, The higher-order Picone identity and comparison of half-linear differential equations of even order, Nonlinear Anal., 74(18), (2011), 7513-7518.
19. [19] J. Jaros, Caccioppoli estimates through an anisotropic Picone's identity, Proc. Amer. Math. Soc., 143(3) (2015), 1137-1144.
20. [20] B.S. Lian, Q.H. Yang, F. Yang, Some weighted Hardy-type inequalities on anisotropic Heisenberg groups, J. Inequal. Appl., 2011, 1-10.
21. [21] H.X. Liu, J.W. Luan, Hardy-type inequalities on a half-space in the Heisenberg group, J. Inequal. Appl., 2013, (2013), 291.
22. [22] A. Manes, A.M. Micheletti, Un'estensione della teoria variazionale classica degli autovalori per operatoriellittici del secondo ordine, Bollettino U.M.I. 7, (1973), 285-301.
23. [23] P. Niu, H. Zhang, Y. Wang, Hardy type and Rellich type inequalities on the Heisenberg group, Proc. Amer. Math. Soc. 129(12) (2001), 3623-3630.
24. [24] P. Niu, H. Zhang and X. Luo, Hardy’s inequalities and Pohozaev’s identities on the Heisenberg group, Acta Math. Sinica. 46(2) (2003), 279-290.
25. [25] M. Picone, Un teorema sulle soluzioni delle equazioni lineari ellittiche autoaggiunte alle derivate parziali del secondo-ordine, Atti Accad. Naz. Lincei Rend. 20 1911, 213-219.
26. [26] J.Tyagi, A nonlinear Picone's identity and applications, Appl. Math. Lett., 26 (2013), 624-626.
27. [27] Y. Xiao, Hardy inequalities with Aharonov-Bohm type magnetic field on the Heisenberg group, J. Inequal. Appl., 2015(1), (2015), 95.
28. [28] N. Yoshida, Picone identities for half-linear elliptic operators with p(x)-Laplacians and applications to Sturmian comparison theory, Nonlinear Anal. 74 (2011), no. 16, 5631-5642.