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Singular double-phase systems with variable growth for the Baouendi-Grushin operator
1. | Mathematics Department, University of Monastir, Faculty of Sciences, 5019 Monastir, Tunisia |
2. | Faculty of Applied Mathematics, AGH University of Science and Technology, 30-059 Kraków, Poland, Department of Mathematics, University of Craiova, 200585 Craiova, Romania |
3. | Institute of Mathematics 'Simion Stoilow' of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania |
In this paper we study a class of singular systems with double-phase energy. The main feature is that the associated Euler equation is driven by the Baouendi-Grushin operator with variable coefficient. In such a way, we continue the analysis introduced in [
References:
[1] |
B. Abdellaoui and I. Peral,
On quasilinear elliptic equations related to some Caffarelli-Kohn-Nirenberg inequalities, Commun. Pure Appl. Anal., 2 (2003), 539-566.
doi: 10.3934/cpaa.2003.2.539. |
[2] |
Adimurthi, N. Chaudhuri and M. Ramaswamy,
An improved Hardy-Sobolev inequality and its applications, Proc. Amer. Math. Soc., 130 (2002), 489-505.
doi: 10.1090/S0002-9939-01-06132-9. |
[3] |
V. Ambrosio and V. D. Rădulescu,
Fractional double-phase patterns: Concentration and multiplicity of solutions, J. Math. Pures Appl., 142 (2020), 101-145.
doi: 10.1016/j.matpur.2020.08.011. |
[4] |
A. Bahrouni and D. D. Repovš,
Existence and nonexistence of solutions for $p(x)$-curl systems arising in electromagnetism, Complex Var. Elliptic Equ., 63 (2018), 292-301.
doi: 10.1080/17476933.2017.1304390. |
[5] |
A. Bahrouni, V. D. Rădulescu and D. D. Repovš,
A weighted anisotropic variant of the Caffarelli-Kohn-Nirenberg inequality and applications, Nonlinearity, 31 (2018), 1516-1534.
doi: 10.1088/1361-6544/aaa5dd. |
[6] |
A. Bahrouni, V. D. Rădulescu and D. D. Repovš,
Double-phase transonic flow problems with variable growth: nonlinear patterns and stationary waves, Nonlinearity, 32 (2019), 2481-2495.
doi: 10.1088/1361-6544/ab0b03. |
[7] |
A. Bahrouni, V. D. Rădulescu and P. Winkert, Double-phase problems with variable growth and convection for the Baouendi-Grushin operator,, Z. Angew. Math. Phys., 71 (2020), Paper No. 183, 15 pp.
doi: 10.1007/s00033-020-01412-7. |
[8] |
P. Baroni, M. Colombo and G. Mingione,
Nonautonomous functionals, borderline cases and related function classes, St. Petersburg Math. J., 27 (2016), 347-379.
doi: 10.1090/spmj/1392. |
[9] |
M. S. Baouendi,
Sur une classe d'opérateurs elliptiques dégénérés, Bull. Soc. Math. France, 95 (1967), 45-87.
|
[10] |
L. Beck and G. Mingione,
Lipschitz bounds and non-uniform ellipticity, Comm. Pure Appl. Math., 73 (2020), 944-1034.
doi: 10.1002/cpa.21880. |
[11] |
L. Caffarelli, R. Kohn and L. Nirenberg,
First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275.
|
[12] |
M. Cencelj, V. D. Rădulescu and D. D. Repovš,
Double-phase problems with variable growth, Nonlinear Anal, 177 (2018), 270-287.
doi: 10.1016/j.na.2018.03.016. |
[13] |
F. Colasuonno and M. Squassina,
Eigenvalues for double-phase variational integrals, Ann. Mat. Pura Appl., 195 (2016), 1917-1959.
doi: 10.1007/s10231-015-0542-7. |
[14] |
F. Colasuonno and P. Pucci,
Multiplicity of solutions for $p(x)$-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal., 74 (2011), 5962-5974.
doi: 10.1016/j.na.2011.05.073. |
[15] |
M. Colombo and G. Mingione,
Bounded minimisers of double-phase variational integrals, Arch. Ration. Mech. Anal., 218 (2015), 219-273.
doi: 10.1007/s00205-015-0859-9. |
[16] |
B. Franchi and M. C. Tesi,
A finite element approximation for a class of degenerate elliptic equations, Math. Comp., 69 (2000), 41-63.
doi: 10.1090/S0025-5718-99-01075-3. |
[17] |
V. V. Grushin,
A certain class of hypoelliptic operators, Mat. Sb. (N.S.), 83 (1970), 456-473.
|
[18] |
P. Hájek, V. Montesinos Santalucía, J. Vanderwerff and V. Zizler, Biorthogonal Systems in Banach Spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 26, Springer, New York, 2008. |
[19] |
W. Liu and G. Dai,
Existence and multiplicity results for double-phase problems, J. Differential Equations, 265 (2018), 4311-4334.
doi: 10.1016/j.jde.2018.06.006. |
[20] |
J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983.
doi: 10.1007/BFb0072210. |
[21] |
N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš,
Nonlinear singular problems with indefinite potential term, Anal. Math. Phys., 9 (2019), 2237-2262.
doi: 10.1007/s13324-019-00333-7. |
[22] |
N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, Double-phase problems with reaction of arbitrary growth,, Z. Angew. Math. Phys., 69 (2018), Art. 108, 21 pp.
doi: 10.1007/s00033-018-1001-2. |
[23] |
N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš,
Double-phase problems and a discontinuity property of the spectrum, Proceedings Amer. Math. Soc., 147 (2019), 2899-2910.
doi: 10.1090/proc/14466. |
[24] |
N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš,
Existence and multiplicity of solutions for double-phase Robin problems, Bull. Lond. Math. Soc., 52 (2020), 546-560.
doi: 10.1112/blms.12347. |
[25] |
N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, Ground state and nodal solutions for a class of double phase problems, Z. Angew. Math. Phys., 71 (2020), Paper No. 15, 15 pp.
doi: 10.1007/s00033-019-1239-3. |
[26] |
N. S. Papageorgiou, C. Vetro and F. Vetro, Positive solutions for singular $(p, 2)$-equations,, Z. Angew. Math. Phys., 70 (2019), Paper No. 72, 10 pp.
doi: 10.1007/s00033-019-1117-z. |
[27] |
V. D. Rădulescu,
Nonlinear elliptic equations with variable exponent: old and new, Nonlinear Anal., 121 (2015), 336-369.
doi: 10.1016/j.na.2014.11.007. |
[28] |
V. D. Rădulescu,
Isotropic and anisotropic double-phase problems: Old and new, Opuscula Mathematica, 39 (2019), 259-279.
doi: 10.7494/OpMath.2019.39.2.259. |
[29] |
V. D. Rădulescu and D. D. Repovš, Partial Differential Equations with Variable Exponents, CRC Press, Boca Raton, FL, 2015.
doi: 10.1201/b18601.![]() ![]() |
[30] |
Q. Zhang and V. D. Rădulescu,
Double-phase anisotropic variational problems and combined effects of reaction and absorption terms, J. Math. Pures Appl., 118 (2018), 159-203.
doi: 10.1016/j.matpur.2018.06.015. |
show all references
References:
[1] |
B. Abdellaoui and I. Peral,
On quasilinear elliptic equations related to some Caffarelli-Kohn-Nirenberg inequalities, Commun. Pure Appl. Anal., 2 (2003), 539-566.
doi: 10.3934/cpaa.2003.2.539. |
[2] |
Adimurthi, N. Chaudhuri and M. Ramaswamy,
An improved Hardy-Sobolev inequality and its applications, Proc. Amer. Math. Soc., 130 (2002), 489-505.
doi: 10.1090/S0002-9939-01-06132-9. |
[3] |
V. Ambrosio and V. D. Rădulescu,
Fractional double-phase patterns: Concentration and multiplicity of solutions, J. Math. Pures Appl., 142 (2020), 101-145.
doi: 10.1016/j.matpur.2020.08.011. |
[4] |
A. Bahrouni and D. D. Repovš,
Existence and nonexistence of solutions for $p(x)$-curl systems arising in electromagnetism, Complex Var. Elliptic Equ., 63 (2018), 292-301.
doi: 10.1080/17476933.2017.1304390. |
[5] |
A. Bahrouni, V. D. Rădulescu and D. D. Repovš,
A weighted anisotropic variant of the Caffarelli-Kohn-Nirenberg inequality and applications, Nonlinearity, 31 (2018), 1516-1534.
doi: 10.1088/1361-6544/aaa5dd. |
[6] |
A. Bahrouni, V. D. Rădulescu and D. D. Repovš,
Double-phase transonic flow problems with variable growth: nonlinear patterns and stationary waves, Nonlinearity, 32 (2019), 2481-2495.
doi: 10.1088/1361-6544/ab0b03. |
[7] |
A. Bahrouni, V. D. Rădulescu and P. Winkert, Double-phase problems with variable growth and convection for the Baouendi-Grushin operator,, Z. Angew. Math. Phys., 71 (2020), Paper No. 183, 15 pp.
doi: 10.1007/s00033-020-01412-7. |
[8] |
P. Baroni, M. Colombo and G. Mingione,
Nonautonomous functionals, borderline cases and related function classes, St. Petersburg Math. J., 27 (2016), 347-379.
doi: 10.1090/spmj/1392. |
[9] |
M. S. Baouendi,
Sur une classe d'opérateurs elliptiques dégénérés, Bull. Soc. Math. France, 95 (1967), 45-87.
|
[10] |
L. Beck and G. Mingione,
Lipschitz bounds and non-uniform ellipticity, Comm. Pure Appl. Math., 73 (2020), 944-1034.
doi: 10.1002/cpa.21880. |
[11] |
L. Caffarelli, R. Kohn and L. Nirenberg,
First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275.
|
[12] |
M. Cencelj, V. D. Rădulescu and D. D. Repovš,
Double-phase problems with variable growth, Nonlinear Anal, 177 (2018), 270-287.
doi: 10.1016/j.na.2018.03.016. |
[13] |
F. Colasuonno and M. Squassina,
Eigenvalues for double-phase variational integrals, Ann. Mat. Pura Appl., 195 (2016), 1917-1959.
doi: 10.1007/s10231-015-0542-7. |
[14] |
F. Colasuonno and P. Pucci,
Multiplicity of solutions for $p(x)$-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal., 74 (2011), 5962-5974.
doi: 10.1016/j.na.2011.05.073. |
[15] |
M. Colombo and G. Mingione,
Bounded minimisers of double-phase variational integrals, Arch. Ration. Mech. Anal., 218 (2015), 219-273.
doi: 10.1007/s00205-015-0859-9. |
[16] |
B. Franchi and M. C. Tesi,
A finite element approximation for a class of degenerate elliptic equations, Math. Comp., 69 (2000), 41-63.
doi: 10.1090/S0025-5718-99-01075-3. |
[17] |
V. V. Grushin,
A certain class of hypoelliptic operators, Mat. Sb. (N.S.), 83 (1970), 456-473.
|
[18] |
P. Hájek, V. Montesinos Santalucía, J. Vanderwerff and V. Zizler, Biorthogonal Systems in Banach Spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 26, Springer, New York, 2008. |
[19] |
W. Liu and G. Dai,
Existence and multiplicity results for double-phase problems, J. Differential Equations, 265 (2018), 4311-4334.
doi: 10.1016/j.jde.2018.06.006. |
[20] |
J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983.
doi: 10.1007/BFb0072210. |
[21] |
N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš,
Nonlinear singular problems with indefinite potential term, Anal. Math. Phys., 9 (2019), 2237-2262.
doi: 10.1007/s13324-019-00333-7. |
[22] |
N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, Double-phase problems with reaction of arbitrary growth,, Z. Angew. Math. Phys., 69 (2018), Art. 108, 21 pp.
doi: 10.1007/s00033-018-1001-2. |
[23] |
N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš,
Double-phase problems and a discontinuity property of the spectrum, Proceedings Amer. Math. Soc., 147 (2019), 2899-2910.
doi: 10.1090/proc/14466. |
[24] |
N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš,
Existence and multiplicity of solutions for double-phase Robin problems, Bull. Lond. Math. Soc., 52 (2020), 546-560.
doi: 10.1112/blms.12347. |
[25] |
N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, Ground state and nodal solutions for a class of double phase problems, Z. Angew. Math. Phys., 71 (2020), Paper No. 15, 15 pp.
doi: 10.1007/s00033-019-1239-3. |
[26] |
N. S. Papageorgiou, C. Vetro and F. Vetro, Positive solutions for singular $(p, 2)$-equations,, Z. Angew. Math. Phys., 70 (2019), Paper No. 72, 10 pp.
doi: 10.1007/s00033-019-1117-z. |
[27] |
V. D. Rădulescu,
Nonlinear elliptic equations with variable exponent: old and new, Nonlinear Anal., 121 (2015), 336-369.
doi: 10.1016/j.na.2014.11.007. |
[28] |
V. D. Rădulescu,
Isotropic and anisotropic double-phase problems: Old and new, Opuscula Mathematica, 39 (2019), 259-279.
doi: 10.7494/OpMath.2019.39.2.259. |
[29] |
V. D. Rădulescu and D. D. Repovš, Partial Differential Equations with Variable Exponents, CRC Press, Boca Raton, FL, 2015.
doi: 10.1201/b18601.![]() ![]() |
[30] |
Q. Zhang and V. D. Rădulescu,
Double-phase anisotropic variational problems and combined effects of reaction and absorption terms, J. Math. Pures Appl., 118 (2018), 159-203.
doi: 10.1016/j.matpur.2018.06.015. |
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