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Comparison Results for Nonlinear Equations Involving A-Harmonic Operator

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Abstract

In this paper, we derive comparison theorems for a pair of nonlinear equations involving A-harmonic operator. In the special cases, our results give the well-known Leighton-type comparison results concerning the Finsler p-Laplacian, the p-Laplacian, and the pseudo p-Laplacian equations.

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References

  1. Abdellaoui, B., Peral, I.: Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential. Ann. Mat. Pura Appl. (4) 182(3), 247–270 (2003)

  2. Allegretto W., Huang Y.X. (1998) A Picone’s identity for the p-Laplacian and applications. Nonlinear Anal. 32(7): 819–830

    Article  MathSciNet  MATH  Google Scholar 

  3. Alvino A., Ferone V., Trombetti G., Lions P.L. (1997) Convex symmetrization and applications (English. with English and French summaries). Ann. Inst. H. Poincaré Anal. Non Linéaire 14(2): 275–293

    Article  MathSciNet  MATH  Google Scholar 

  4. Bellettini G., Paolini M. (1996) Anisotropic motion by mean curvature in the context of Finsler geometry. Hokkaido Math. J. 25(3): 537–566

    Article  MathSciNet  MATH  Google Scholar 

  5. Berestycki H., Capuzzo-Dolcetta I., Nirenberg L. (1995) Variational methods for indefinite superlinear homogeneous elliptic problems. NoDEA Nonlinear Differ. Equ. Appl. 2: 553–572

    Article  MathSciNet  MATH  Google Scholar 

  6. Bognár G., Došlý O. (2003) The application of Picone-type identity for some nonlinear elliptic differential equations. Acta Math. Univ. Comen. New Ser. 72: 45–57

    MathSciNet  MATH  Google Scholar 

  7. Cianchi A., Salani P. (2009) Overdetermined anisotropic elliptic problems. Math. Ann. 345(4): 859–881

    Article  MathSciNet  MATH  Google Scholar 

  8. Cuesta M., Quoirin H.R. (2009) A weighted eigenvalue problem for the p-Laplacain plus a potential. Nonlinear Differ. Equ. Appl. 16: 469–491

    Article  MATH  Google Scholar 

  9. Della Pietra F., Gavitone N. (2013) Anisotropic elliptic problems involving Hardy-type potentials. J. Math. Anal. Appl. 397: 800–813

    Article  MathSciNet  MATH  Google Scholar 

  10. Del Pezzo L.M., Bonder J.F. (2006) An optimization problem for the first eigenvalue of the p-Laplacian plus a potential. Commun. Pure Appl. Anal. 5: 675–690

    Article  MathSciNet  MATH  Google Scholar 

  11. Došlý O., Mařík R. (2001) Nonexistence of positive solutions of PDE’s with p-Laplacian. Acta. Math. Hung. 90: 89–107

    Article  MATH  Google Scholar 

  12. Dunninger, D.R.: A Sturm comparison theorem for some degenerate quasilinear elliptic operators (English, with Italian summary). Boll. Un. Mat. Ital. A (7) 9(1), 117–121 (1995)

  13. Ferone V., Kawohl B. (2009) Remarks on a Finsler-Laplacian. Proc. Am. Math. Soc. 137(1): 247–253

    Article  MathSciNet  MATH  Google Scholar 

  14. Jaroš J. (2014) Comparison results for nonlinear elliptic equations involving a Finsler-Laplacian. Acta. Math. Univ. Comen. LXXXIII: 81–91

    MathSciNet  MATH  Google Scholar 

  15. Jaroš J. (2014) Picone’s identity for a Finsler p-Laplacian and comparison of nonlinear elliptic equations. Math. Bohem. 139(3): 535–552

    MathSciNet  MATH  Google Scholar 

  16. Jaroš J. (2015) A-harmonic Picone’s identity with applications. Ann. Math. 194: 719–729

    MATH  Google Scholar 

  17. Jaroš J. (2015) Caccioppoli estimates through an anisotropic Picone’s identity. Proc. Am. Math. Soc. 143(3): 1137–1144

    MATH  Google Scholar 

  18. Kusano T., Jaroš J., Yoshida N. (2000) A Picone-type identity and Sturmian comparison and oscillation theorems for a class of half-linear partial differential equations of second order. Nonlinear Anal. Theory Methods Appl. Ser. A Theory Methods 40: 381–395

    Article  MathSciNet  MATH  Google Scholar 

  19. Liskevich V., Lyakhova S., Moroz V. (2007) Positive solutions to nonlinear p-Laplace equations with Hardy potential in exterior domains. J. Differ. Equ. 232(1): 212–252

    Article  MathSciNet  MATH  Google Scholar 

  20. Lucia M., Prashanth S. (2007) Simplicity of principal eigenvalue for p-Laplace operator with singular indefinite weight. Arch. Math. 86: 79–89

    Article  MathSciNet  MATH  Google Scholar 

  21. Picone M. (1911) Un teorema sulle soluzioni delle equazioni lineari ellittiche autoaggiunte alle derivate parziali del secondo-ordine. Atti Accad. Naz. Lincei Rend. 20: 213–219

    MATH  Google Scholar 

  22. Swanson, C.A.: Picone’s identity (English, with Italian summary). Rend. Mat. (6) 8(2), 373–397 (1975). Collection of articles dedicated to Mauro Picone on the occasion of his ninetieth birthday, II

  23. Swanson, C.A.: Comparison and Oscillation Theory of Linear Differential Equations. Mathematics in Science and Engineering, vol. 48. Academic Press, New York (1968)

  24. Wang G., Xia C. (2011) A characterization of the Wulff shape by an overdetermined anisotropic PDE. Arch. Ration. Mech. Anal. 199(1): 99–115

    Article  MathSciNet  MATH  Google Scholar 

  25. Wang G., Xia C. (2012) An optimal anisotropic Poincaré inequality for convex domains. Pac. J. Math. 258(2): 305–326

    Article  MathSciNet  MATH  Google Scholar 

  26. Wang G., Xia C. (2012) Blow-up analysis of a Finsler–Liouville equation in two dimensions. J. Differ. Equ. 252(2): 1668–1700

    Article  MathSciNet  MATH  Google Scholar 

  27. Yoshida, N.: Oscillation Theory of Partial Differential Equations. World Scientific, Hackensack (2008)

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Gazi University, 06500, Ankara, Turkey

    Adil Misir

  2. Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Izmir University, 35350, Uckuyular, Izmir, Turkey

    Aydın Tiryaki

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  1. Adil Misir
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  2. Aydın Tiryaki
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Correspondence to Adil Misir.

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Misir, A., Tiryaki, A. Comparison Results for Nonlinear Equations Involving A-Harmonic Operator. Mediterr. J. Math. 13, 4949–4959 (2016). https://doi.org/10.1007/s00009-016-0785-6

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  • DOI: https://doi.org/10.1007/s00009-016-0785-6

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