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

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

* Corresponding author: Vicenţiu D. Rădulescu

Received  November 2020 Published  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.

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]

AdimurthiN. 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. BahrouniV. 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. BahrouniV. 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. BaroniM. 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. CaffarelliR. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275.   Google Scholar

[12]

M. CenceljV. 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. PapageorgiouV. 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. PapageorgiouV. 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. PapageorgiouV. 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]

AdimurthiN. 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. BahrouniV. 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. BahrouniV. 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. BaroniM. 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. CaffarelliR. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275.   Google Scholar

[12]

M. CenceljV. 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. PapageorgiouV. 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. PapageorgiouV. 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. PapageorgiouV. 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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