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September  2021, 41(9): 4283-4296. doi: 10.3934/dcds.2021036

Singular double-phase systems with variable growth for the Baouendi-Grushin operator

Anouar Bahrouni 1, and  Vicenţiu D. Rădulescu 2,3,,

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

* Corresponding author: Vicenţiu D. Rădulescu

Received  November 2020 Published  September 2021 Early access  February 2021

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 [6] to the case of lack of compactness corresponding to the whole Euclidean space. After establishing a related compactness property, we establish the existence of solutions for the Baouendi-Grushin singular system.

Keywords: Baouendi-Grushin operator, double-phase problem, singular term, degenerate differential operator, existence of solutions.
Mathematics Subject Classification: Primary: 35J70, 35P30; Secondary: 76H05.
Citation: Anouar Bahrouni, Vicenţiu D. Rădulescu. Singular double-phase systems with variable growth for the Baouendi-Grushin operator. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4283-4296. doi: 10.3934/dcds.2021036
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

M. S. Baouendi, Sur une classe d'opérateurs elliptiques dégénérés, Bull. Soc. Math. France, 95 (1967), 45-87.   Google Scholar

[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.  Google Scholar

[11]

L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

V. V. Grushin, A certain class of hypoelliptic operators, Mat. Sb. (N.S.), 83 (1970), 456-473.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983. doi: 10.1007/BFb0072210.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29] V. D. Rădulescu and D. D. Repovš, Partial Differential Equations with Variable Exponents, CRC Press, Boca Raton, FL, 2015.  doi: 10.1201/b18601.  Google Scholar
[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

M. S. Baouendi, Sur une classe d'opérateurs elliptiques dégénérés, Bull. Soc. Math. France, 95 (1967), 45-87.   Google Scholar

[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.  Google Scholar

[11]

L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

V. V. Grushin, A certain class of hypoelliptic operators, Mat. Sb. (N.S.), 83 (1970), 456-473.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983. doi: 10.1007/BFb0072210.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29] V. D. Rădulescu and D. D. Repovš, Partial Differential Equations with Variable Exponents, CRC Press, Boca Raton, FL, 2015.  doi: 10.1201/b18601.  Google Scholar
[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.  Google Scholar

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