Picone’s identity for Δγ-Laplace operator and its applications
DOI:
https://doi.org/10.37863/umzh.v73i4.639Keywords:
Δγ-Laplace operator, Picone’s identit, Sturmian comparison theore, Monotonicity of the eigenvalu, Hardy’s inequalitAbstract
UDC 517.9
We prove a nonlinear analogue of Picone's identity for Δγ-Laplace operator.
As an application, we give a Hardy type inequality and Sturmian comparison principle.
We also show the strict monotonicity of the principle eigenvalue and degenerate elliptic system.
References
W. Allegretto, Positive solutions and spectral properties of weakly coupled elliptic systems, J. Math. Anal. and Appl., 120, № 2, 723 – 729 (1986), https://doi.org/10.1016/0022-247X(86)90191-5 DOI: https://doi.org/10.1016/0022-247X(86)90191-5
W. Allegretto, On the principal eigenvalues of indefinite elliptic problems, Math. Z., 195, № 1, 29 – 35 (1987), https://doi.org/10.1007/BF01161596 DOI: https://doi.org/10.1007/BF01161596
W. Allegretto, Sturmian theorems for second order systems, Proc. Amer. Math. Soc., 94, № 2, 291 – 296 (1985), https://doi.org/10.2307/2045393 DOI: https://doi.org/10.1090/S0002-9939-1985-0784181-8
W. Allegretto, Y. X. Huang, A Picone’s identity for the p-Laplacian and applications, Nonlinear Anal., 32, № 7, 819 – 830 (1998), https://doi.org/10.1016/S0362-546X(97)00530-0 DOI: https://doi.org/10.1016/S0362-546X(97)00530-0
C. T. Anh, B. K. My, Existence of solutions to Deltalambda -Laplace equations without the Ambrosetti – Rabinowitz condition, Complex Var. and Elliptic Equat., 61, № 1, 137 – 150 (2016), https://doi.org/10.1080/17476933.2015.1068762 DOI: https://doi.org/10.1080/17476933.2015.1068762
K. Bal, Generalized Picone’s identity and its applications, Electron. J. Different. Equat., № 243 (2013), 6 p.
B. Franchi, E. Lanconelli, A metric associated with a class of degenerate elliptic operators, Conf. Linear Partial and Pseudodifferential Operators (Torino, 1982), Rend. Sem. Mat. Univ. Politec. Torino, 1983, Special Issue, 105 – 114 (1984).
B. Franchi, E. Lanconelli, An embedding theorem for Sobolev spaces related to nonsmooth vector fields and Harnack inequality, Commun. Part. Different. Equat., 9, № 13, 1237 – 1264 (1984), https://doi.org/10.1080/03605308408820362 DOI: https://doi.org/10.1080/03605308408820362
V. V. Grushin, A certain class of hypoelliptic operators, Mat. Sb. (N.S.), 83, № 125, 456 – 473 (1970) (in Russian).
A. E. Kogoj, E. Lanconelli, On semilinear Deltalambda -Laplace equation, Nonlinear Anal., 75, № 12, 4637 – 4649 (2012), https://doi.org/10.1016/j.na.2011.10.007 DOI: https://doi.org/10.1016/j.na.2011.10.007
A. E. Kogoj, S. Sonner, Attractors for a class of semi-linear degenerate parabolic equations, J. Evol. Equat., 13, № 3, 675 – 691 (2013), https://doi.org/10.1007/s00028-013-0196-0 DOI: https://doi.org/10.1007/s00028-013-0196-0
D. T. Luyen, D. T. Huong, L. T. H. Hanh, Existence of infinitely many solutions for Deltagamma -Laplace problems, Math. Notes, 103, № 5, 724 – 736 (2018), https://doi.org/10.1134/S000143461805005X DOI: https://doi.org/10.1134/S000143461805005X
D. T. Luyen, Two nontrivial solutions of boundary-value problems for semilinear Deltagamma -differential equations, Math. Notes, 101, № 5, 815 – 823 (2017), https://doi.org/10.1134/S0001434617050078 DOI: https://doi.org/10.1134/S0001434617050078
D. T. Luyen, Existence of nontrivial solution for fourth-order semilinear Deltagamma-Laplace equation in BbbRN , Electron. J. Qual. Theory Different. Equat., 78, 1 – 12 (2019), https://doi.org/10.14232/ejqtde.2019.1.78 DOI: https://doi.org/10.14232/ejqtde.2019.1.78
D. T. Luyen, Multiple solutions for semilinear Deltagamma- differential equations in BbbRN with sign-changing potential, Commun. Math. Anal., 22, № 1, 61 – 75 (2019).
D. T. Luyen, L. T. H. Hanh, Three nontrivial solutions of boundary-value problems for semilinear Deltagamma-Laplace equation, Bol. Soc. Parana. Mat. https://doi:10.5269/bspm.45841 (2019).
D. T. Luyen, N. M. Tri, Existence of solutions to boundary-value problems for semilinear Deltagamma -differential equations, Math. Notes, 97, № 1, 73 – 84 (2015), https://doi.org/10.1134/S0001434615010101 DOI: https://doi.org/10.1134/S0001434615010101
D. T. Luyen, N. M. Tri, Large-time behavior of solutions to damped hyperbolic equation involving strongly degenerate elliptic differential operators, Siberian Math. J., 57, № 4, 632 – 649 (2016), https://doi.org/10.1134/s0037446616040078 DOI: https://doi.org/10.1134/S0037446616040078
D. T. Luyen, N. M. Tri, Global attractor of the Cauchy problem for a semilinear degenerate damped hyperbolic equation involving the Grushin operator, Ann. Polon. Math., 117, № 2, 141 – 162 (2016), https://doi.org/10.4064/ap3831-3-2016 DOI: https://doi.org/10.4064/ap3831-3-2016
D. T. Luyen, N. M. Tri, Existence of infinitely many solutions for semilinear degenerate Schr¨odinger equations, J. Math. Anal. and Appl., 461, № 2, 1271 – 1286 (2018), https://doi.org/10.1016/j.jmaa.2018.01.016 DOI: https://doi.org/10.1016/j.jmaa.2018.01.016
D. T. Luyen, N. M. Tri, On the existence of multiple solutions to boundary value problems for semilinear elliptic degenerate operators, Complex Var. and Elliptic Equat., 64, № 6, 1050 – 1066 (2019), https://doi.org/10.1080/17476933.2018.1498086 DOI: https://doi.org/10.1080/17476933.2018.1498086
A. Manes, A. M. Micheletti, Un’estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine, Boll. Unione Mat. Ital., 7, № 4, 285 – 301 (1973) (in Italian).
B. Rahal, M. K. Hamdani, Infinitely many solutions for Deltaalpha -Laplace equations with sign-changing potential, J. Fixed Point Theory and Appl., 20, № 4 (2018), https://doi.org/10.1007/s11784-018-0617-3 DOI: https://doi.org/10.1007/s11784-018-0617-3
P. T. Thuy, N. M. Tri, Nontrivial solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations, Nonlinear Different. Equat. and Appl., 19, № 3, 279 – 298 (2012), https://doi.org/10.1007/s00030-011-0128-z DOI: https://doi.org/10.1007/s00030-011-0128-z
P. T. Thuy, N. M. Tri, Long time behavior of solutions to semilinear parabolic equations involving strongly degenerate elliptic differential operators, Nonlinear Different. Equat. and Appl., 20, № 3, 1213 – 1224 (2013), https://doi.org/10.1007/s00030-012-0205-y DOI: https://doi.org/10.1007/s00030-012-0205-y
N. M. Tri, Critical Sobolev exponent for hypoelliptic operators, Acta Math. Vietnam, 23, № 1, 83 – 94 (1998).
N. M. Tri, Semilinear degenerate elliptic differential equations, local and global theories, Lambert Acad. Publ. (2010).
N. M. Tri, Recent progress in the theory of semilinear equations involving degenerate elliptic differential operators, Publ. House Sci. and Technology Vietnam Acad. Sci. and Technology (2014).
J. Tyagi, A nonlinear Picone’s identity and its applications, Appl. Math. Lett., 26, № 6, 624 – 626 (2013), https://doi.org/10.1016/j.aml.2012.12.020 DOI: https://doi.org/10.1016/j.aml.2012.12.020
