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

Anouar Bahrouni; Vicenţiu D. Rădulescu

doi:10.3934/dcds.2021036

Article Contents

2021, Volume 41, Issue 9: 4283-4296. Doi: 10.3934/dcds.2021036

This issue Previous Article Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues Next Article Supercritical elliptic problems involving a Cordes like operator

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
^* Corresponding author: Vicenţiu D. Rădulescu

Received: November 2020

Early access: February 2021

Published: September 2021

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Abstract

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.

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Mathematics Subject Classification: Primary: 35J70, 35P30; Secondary: 76H05.

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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.