Comparison results for nonlinear elliptic equations involving a Finsler-Laplacian
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Abstract
Picone identity for a Finsler-Lapace operator is established and comparison theorems of the Leighton type for a pair of nonlinear elliptic equations involving such operators are obtained with the help of this new formula.
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Jaroš, J.
(2014).
Comparison results for nonlinear elliptic equations involving a Finsler-Laplacian.
Acta Mathematica Universitatis Comenianae, 83(1), 81-91.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/79/30
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References
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[2] W. Allegretto, Sturm type theorems for solutions of elliptic nonlinear problems, NoDEA Nonlinear Dierential Equations Appl. 7 (2000) 309-321.
[3] A. Alvino, V. Ferone, G. Trombetti and P.-L. Lions, Convex symmetrization and applications, Ann. Inst. Henri Poincare Anal. Non Lineaire 14 (1997) 275-293.
[4] M. Belloni, V. Ferone, B. Kawohl, Isoperimetric inequalities, Wul shape and related questions for strongly nonlinear elliptic operators, Z. Angew. Math. Phys. (ZAMP) 41 (2004) 439-505.
[5] G. Bellettini, M. Paolini, Anisotropic motion by mean curvature in the context of Finsler geometry, Hokkaido Math. J. 25 (1996) 537-566.
[6] H. Berestycki, I. Capuzzo-Dolcetta, L. Nirenberg, Variational methods for indenite superlinear homogeneous elliptic problems, NoDEA Nonlinear Differential Equations Appl. 2 (1995) 553-572.
[7] G. Bognar, O. Dosly, The application of Picone-type identity for some nonlinear elliptic dierential equations, Acta Math. Univ. Comenian 72 (2003) 45-57.
[8] A. Cianchi, P. Salani, Overdetermined anisotropic elliptic problems, Math. Ann. 345 (2009) 859-881.
[9] F. Della Pietra, N. Gavitone, Anisotropic elliptic problems involving Hardy-type potentials, J. Math. Anal. Appl. 397 (2013) 800-813.
[10] O. Dosly, The Picone identity for a class of partial dierential equations, Math. Bohem. 127 (2002) 581-589.
[11] D. R. Dunninger, A Sturm comparison theorem for some degenerate quasilinear elliptic operators, Boll. Unione Mat. Ital. A (7) 9 (1995), 117-121.
[12] V. Ferone, B. Kawohl, Remarks on a Finsler-Laplacian, Proc. Amer. Math. Soc. 137 (2009) 247-253.
[13] T. Kusano, J. Jaros, N. Yoshida, A Picone-type identity and Sturmian comparison and oscillation theorems for a class of half-linear partial dierential equations of second order, Nonlinear Anal. 40 (2000) 381-395.
[14] 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.
[15] W. T. Reid, Sturmian Theory for Ordinary Dierential Equations, Springer-Verlag, New York, Heidelberg, Berlin, 1980.
[16] C. A. Swanson, Picone's identity, Rend. Mat 8 (1975) 373-397.
[17] G. Wang, C. Xia, A characterization of the Wul shape by an overdetermined anisotropic PDE, Arch. Ration. Mech. Anal. 199 (2011) 99-115.
[18] G. Wang, C. Xia, An optimal anisotropic Poincare inequality for convex domains, Pac. J. Math. 258 (2012) 305-326.
[19] G. Wang. C. Xia, Blow-up analysis of a Finsler-Liouville equation in two dimensions, J. Differential Equations 252 (2012) 1668-1700.
[20] N. Yoshida, Oscillation Theory of Partial Differential Equations, World Scientic, Singapore, Hackensack, London, 2008.
[2] W. Allegretto, Sturm type theorems for solutions of elliptic nonlinear problems, NoDEA Nonlinear Dierential Equations Appl. 7 (2000) 309-321.
[3] A. Alvino, V. Ferone, G. Trombetti and P.-L. Lions, Convex symmetrization and applications, Ann. Inst. Henri Poincare Anal. Non Lineaire 14 (1997) 275-293.
[4] M. Belloni, V. Ferone, B. Kawohl, Isoperimetric inequalities, Wul shape and related questions for strongly nonlinear elliptic operators, Z. Angew. Math. Phys. (ZAMP) 41 (2004) 439-505.
[5] G. Bellettini, M. Paolini, Anisotropic motion by mean curvature in the context of Finsler geometry, Hokkaido Math. J. 25 (1996) 537-566.
[6] H. Berestycki, I. Capuzzo-Dolcetta, L. Nirenberg, Variational methods for indenite superlinear homogeneous elliptic problems, NoDEA Nonlinear Differential Equations Appl. 2 (1995) 553-572.
[7] G. Bognar, O. Dosly, The application of Picone-type identity for some nonlinear elliptic dierential equations, Acta Math. Univ. Comenian 72 (2003) 45-57.
[8] A. Cianchi, P. Salani, Overdetermined anisotropic elliptic problems, Math. Ann. 345 (2009) 859-881.
[9] F. Della Pietra, N. Gavitone, Anisotropic elliptic problems involving Hardy-type potentials, J. Math. Anal. Appl. 397 (2013) 800-813.
[10] O. Dosly, The Picone identity for a class of partial dierential equations, Math. Bohem. 127 (2002) 581-589.
[11] D. R. Dunninger, A Sturm comparison theorem for some degenerate quasilinear elliptic operators, Boll. Unione Mat. Ital. A (7) 9 (1995), 117-121.
[12] V. Ferone, B. Kawohl, Remarks on a Finsler-Laplacian, Proc. Amer. Math. Soc. 137 (2009) 247-253.
[13] T. Kusano, J. Jaros, N. Yoshida, A Picone-type identity and Sturmian comparison and oscillation theorems for a class of half-linear partial dierential equations of second order, Nonlinear Anal. 40 (2000) 381-395.
[14] 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.
[15] W. T. Reid, Sturmian Theory for Ordinary Dierential Equations, Springer-Verlag, New York, Heidelberg, Berlin, 1980.
[16] C. A. Swanson, Picone's identity, Rend. Mat 8 (1975) 373-397.
[17] G. Wang, C. Xia, A characterization of the Wul shape by an overdetermined anisotropic PDE, Arch. Ration. Mech. Anal. 199 (2011) 99-115.
[18] G. Wang, C. Xia, An optimal anisotropic Poincare inequality for convex domains, Pac. J. Math. 258 (2012) 305-326.
[19] G. Wang. C. Xia, Blow-up analysis of a Finsler-Liouville equation in two dimensions, J. Differential Equations 252 (2012) 1668-1700.
[20] N. Yoshida, Oscillation Theory of Partial Differential Equations, World Scientic, Singapore, Hackensack, London, 2008.