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# Article

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Keywords:
Picone identity; Finsler \$p\$-Laplacian
Summary:
In the paper we present an identity of the Picone type for a class of nonlinear differential operators of the second order involving an arbitrary norm \$H\$ in \$\mathbb {R}^n\$ which is continuously differentiable for \$x \not = 0\$ and such that \$H^p\$ is strictly convex for some \$p > 1\$. Two important special cases are the \$p\$-Laplacian and the so-called pseudo \$p\$-Laplacian. The identity is then used to establish a variety of comparison results concerning nonlinear degenerate elliptic equations which involve such operators. We also get criteria for the nonexistence of positive solutions in exterior domains for such equations by means of comparison with the equation exhibiting a kind of “anisotropic radial symmetry”.
References:
[1] Allegretto, W., Huang, Y. X.: A Picone's identity for the \$p\$-Laplacian and applications. Nonlinear Anal., Theory Methods Appl. 32 819-830 (1998). MR 1618334 | Zbl 0930.35053
[2] Alvino, A., Ferone, V., Trombetti, G., Lions, P.-L.: Convex symmetrization and applications. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 14 275-293 (1997). DOI 10.1016/S0294-1449(97)80147-3 | MR 1441395 | Zbl 0877.35040
[3] Bellettini, G., Paolini, M.: Anisotropic motion by mean curvature in the context of Finsler geometry. Hokkaido Math. J. 25 537-566 (1996). DOI 10.14492/hokmj/1351516749 | MR 1416006 | Zbl 0873.53011
[4] Belloni, M., Ferone, V., Kawohl, B.: Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators. Z. Angew. Math. Phys. 54 771-783 (2003). DOI 10.1007/s00033-003-3209-y | MR 2019179 | Zbl 1099.35509
[5] Berestycki, H., Capuzzo-Dolcetta, I., Nirenberg, L.: Variational methods for indefinite superlinear homogeneous elliptic problems. NoDEA, Nonlinear Differ. Equ. Appl. 2 553-572 (1995). DOI 10.1007/BF01210623 | MR 1356874 | Zbl 0840.35035
[6] Bognár, G., Došlý, O.: The application of Picone-type identity for some nonlinear elliptic differential equations. Acta Math. Univ. Comen., New Ser. 72 45-57 (2003). MR 2020577 | Zbl 1106.35013
[7] Cianchi, A., Salani, P.: Overdetermined anisotropic elliptic problems. Math. Ann. 345 859-881 (2009). DOI 10.1007/s00208-009-0386-9 | MR 2545870 | Zbl 1179.35107
[8] Pietra, F. Della, Gavitone, N.: Anisotropic elliptic problems involving Hardy-type potentials. J. Math. Anal. Appl. 397 800-813 (2013). DOI 10.1016/j.jmaa.2012.08.008 | MR 2979615
[9] Došlý, O.: The Picone identity for a class of partial differential equations. Math. Bohem. 127 581-589 (2002). MR 1942643 | Zbl 1074.35521
[10] Došlý, O., Maří{k}, R.: Nonexistence of positive solutions of PDE's with \$p\$-Laplacian. Acta Math. Hung. 90 89-107 (2001). DOI 10.1023/A:1006739909182 | MR 1910321 | Zbl 1062.35022
[11] Došlý, O., Řehák, P.: Half-Linear Differential Equations. North-Holland Mathematics Studies 202 Elsevier, Amsterdam (2005). MR 2158903 | Zbl 1090.34001
[12] Dunninger, D. R.: A Sturm comparison theorem for some degenerate quasilinear elliptic operators. Boll. Unione Mat. Ital., VII. Ser., A 9 117-121 (1995). MR 1324611 | Zbl 0834.35011
[13] Ferone, V., Kawohl, B.: Remarks on a Finsler-Laplacian. Proc. Am. Math. Soc. 137 247-253 (2009). DOI 10.1090/S0002-9939-08-09554-3 | MR 2439447 | Zbl 1161.35017
[14] Kusano, T., Jaroš, J., Yoshida, N.: 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 (2000). DOI 10.1016/S0362-546X(00)85023-3 | MR 1768900 | Zbl 0954.35018
[15] Kusano, T., Naito, Y.: Oscillation and nonoscillation criteria for second order quasilinear differential equations. Acta Math. Hung. 76 81-99 (1997). DOI 10.1007/BF02907054 | MR 1459772 | Zbl 0906.34024
[16] Kusano, T., Naito, Y., Ogata, A.: Strong oscillation and nonoscillation of quasilinear differential equations of second order. Differ. Equ. Dyn. Syst. 2 1-10 (1994). MR 1386034
[17] Picone, M.: Un teorema sulle soluzioni delle equazioni lineari ellittiche autoaggiunte alle derivate parziali del secondo ordine. Rom. Acc. L. Rend. (5) 20 213-219 Italian (1911).
[18] Rockafellar, R. T.: Convex Analysis. Reprint of the 1970 original. Princeton Landmarks in Mathematics Princeton University Press, Princeton (1997). MR 1451876
[19] Swanson, C. A.: Picone's identity. Rend. Mat., VI. Ser. 8 373-397 (1975). MR 0402188 | Zbl 0327.34028
[20] Swanson, C. A.: Comparison and Oscillation Theory of Linear Differential Equations. Mathematics in Science and Engineering 48 Academic Press, New York (1968). MR 0463570 | Zbl 0191.09904
[21] Wang, G., Xia, C.: An optimal anisotropic Poincaré inequality for convex domains. Pac. J. Math. 258 305-326 (2012). DOI 10.2140/pjm.2012.258.305 | MR 2981956 | Zbl 1266.35120
[22] Wang, G., Xia, C.: Blow-up analysis of a Finsler-Liouville equation in two dimensions. J. Differ. Equations 252 1668-1700 (2012). DOI 10.1016/j.jde.2011.08.001 | MR 2853556 | Zbl 1233.35053
[23] Wang, G., Xia, C.: A characterization of the Wulff shape by an overdetermined anisotropic PDE. Arch. Ration. Mech. Anal. 199 99-115 (2011). DOI 10.1007/s00205-010-0323-9 | MR 2754338 | Zbl 1232.35103
[24] Yoshida, N.: Oscillation Theory of Partial Differential Equations. World Scientific, Hackensack (2008). MR 2485076 | Zbl 1154.35001

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