Opuscula Math. 40, no. 4 (2020), 483-494
https://doi.org/10.7494/OpMath.2020.40.4.483

Opuscula Mathematica

# Properties of solutions to some weighted p-Laplacian equation

Prashanta Garain

Abstract. In this paper, we prove some qualitative properties for the positive solutions to some degenerate elliptic equation given by $-\text{div}\big(w|\nabla u|^{p-2}\nabla u\big)=f(x,u),\quad w\in \mathcal{A}_p,$ on smooth domain and for varying nonlinearity $$f$$.

Keywords: $$p$$-Laplacian, degenerate elliptic equations, weighted Sobolev space.

Mathematics Subject Classification: 35A01, 35J62, 35J70.

Full text (pdf)

1. W. Allegretto, Y.X. Huang, A Picone's identity for the $$p$$-Laplacian and applications, Nonlinear Anal. 32 (1998) 7, 819-830.
2. K. Bal, Generalized Picone's identity and its applications, Electron. J. Differential Equations 2013 (2013) 243, 1-6.
3. K. Bal, Uniqueness of a positive solution for quasilinear elliptic equations in Heisenberg group, Electron. J. Differential Equations 2016 (2016) 130, 1-7.
4. K. Bal, P. Garain, I. Mandal, Some qualitative properties of Finsler $$p$$-Laplacian, Indag. Math. (N.S.) 28 (2017) 6, 1258-1264.
5. M. Belloni, B. Kawohl, A direct uniqueness proof for equations involving the $$p$$-Laplace operator, Manuscripta Math. 109 (2002) 2, 229-231.
6. S. Chanillo, R.L. Wheeden, Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions, Amer. J. Math. 107 (1985) 5, 1191-1226.
7. V. De Cicco, M.A. Vivaldi, Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions, Adv. Math. Sci. Appl. 9 (1999) 1, 183-207.
8. P. Drábek, A. Kufner, F. Nicolosi, Quasilinear elliptic equations with degenerations and singularities, De Gruyter Series in Nonlinear Analysis and Applications, vol. 5, Walter de Gruyter & Co., Berlin, 1997.
9. G. Dwivedi, J. Tyagi, Some qualitative questions on the equation $$-\operatorname{div}(a(x,u,\nabla u))=f(x,u)$$, J. Math. Anal. Appl. 446 (2017) 1, 456-469.
10. E.B. Fabes, C.E. Kenig, R.P. Serapioni, The local regularity of solutions of degenerated elliptic equations, Commun. Partial Diff. Eq. 7 (1982) 1, 77-116.
11. J. Heinonen, T. Kilpelainen, O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993.
12. J. Jaroš, Picone's identity for the $$p$$-biharmonic operator with applications, Electron. J. Differential Equations 2011 (2011) 122, 1-6.
13. J. Jaroš, Generalized Picone identity and comparison of half-linear differential equations of order 4m, Mem. Differ. Equ. Math. Phys. 57 (2012) 41-50.
14. J. Jaroš, Picone's identity for a system of first-order nonlinear partial differential equations, Electron. J. Differential Equations 2013 (2013) 143, 1-7.
15. J. Jaroš, Picone's identity for a Finsler $$p$$-Laplacian and comparison of nonlinear elliptic equations, Math. Bohem. 139 (2014) 3, 535-552.
16. J. Jaroš, $$A$$-harmonic Picone's identity with applications, Ann. Mat. Pura Appl. (4) 194 (2015) 3, 719-729.
17. J. Jaroš, Caccioppoli estimates through an anisotropic Picone's identity, Proc. Amer. Math. Soc. 143 (2015) 3, 1137-1144.
18. J. Jaroš, K. Takaŝi, N. Yoshida, Picone-type inequalities for nonlinear elliptic equations with first-order terms and their applications, J. Inequal. Appl. (2006), 1-17.
19. B. Kawohl, M. Lucia, S. Prashanth, Simplicity of the principal eigenvalue for indefinite quasilinear problems, Adv. Differential Equations 12 (2007), 407-434.
20. J. Tyagi, A nonlinear Picone's identity and its applications, Appl. Math. Lett. 26 (2013) 6, 624-626.
• Communicated by Patrizia Pucci.
• Revised: 2020-03-20.
• Accepted: 2020-05-18.
• Published online: 2020-07-09.