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December  2021, 14(12): 4465-4502. doi: 10.3934/dcdss.2021111

Anisotropic singular double phase Dirichlet problems

1. 

Department of Mathematics, Zografou Campus, National Technical University, Athens 15780, Greece

2. 

Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Krakow, Poland

3. 

Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13, Craiova 200585, Romania

4. 

School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

* Corresponding author: Youpei Zhang (zhangypzn@163.com; youpei.zhang@inf.ucv.ro)

Received  July 2021 Revised  September 2021 Published  December 2021 Early access  October 2021

We consider an anisotropic double phase problem with a reaction in which we have the competing effects of a parametric singular term and a superlinear perturbation. We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter varies on $ \mathring{\mathbb{R}}_+ = (0, +\infty) $. Our approach uses variational tools together with truncation and comparison techniques as well as several general results of independent interest about anisotropic equations, which are proved in the Appendix.

Citation: Nikolaos S. Papageorgiou, Vicenţiu D. Rǎdulescu, Youpei Zhang. Anisotropic singular double phase Dirichlet problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (12) : 4465-4502. doi: 10.3934/dcdss.2021111
References:
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E. Acerbi and G. Mingione, Gradient estimates for the $p(x)$-Laplacian system, J. Reine Angew. Math., 584 (2005), 117–148. doi: 10.1515/crll.2005.2005.584.117.  Google Scholar

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A. M. Alghamdi, S. Gala, C. Qian and M. A. Ragusa, The anisotropic integrability logarithmic regularity criterion for the 3D MHD equations, Electron. Res. Arch., 28 (2020), 183–193. doi: 10.3934/era.2020012.  Google Scholar

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A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349–381. doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

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A. Bahrouni and V. D. Rădulescu, Singular double-phase systems with variable growth for the Baouendi-Grushin operator, Discrete Contin. Dyn. Syst., 41 (2021), 4283–4296. doi: 10.3934/dcds.2021036.  Google Scholar

[5]

A. Bahrouni, V. D. Rǎdulescu 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. Rǎdulescu 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

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J. M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. Trans. Roy. Soc. London Ser. A, 306 (1982), 557–611. doi: 10.1098/rsta.1982.0095.  Google Scholar

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L. Beck and G. Mingione, Lipschitz bounds and nonuniform ellipticity, Comm. Pure Appl. Math., 73 (2020), 944–1034. doi: 10.1002/cpa.21880.  Google Scholar

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D. Bonheure, P. d'Avenia and A. Pomponio, On the electrostatic Born-Infeld equation with extended charges, Comm. Math. Phys., 346 (2016), 877–906. doi: 10.1007/s00220-016-2586-y.  Google Scholar

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S.-S. Byun and E. Ko, Global $C^{1, \alpha}$ regularity and existence of multiple solutions for singular $p(x)$-Laplacian equations, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 76, 29 pp. doi: 10.1007/s00526-017-1152-6.  Google Scholar

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M. G. Crandall, P. H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations, 2 (1977), 193–222. doi: 10.1080/03605307708820029.  Google Scholar

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L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Math, Vol. 2017, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.  Google Scholar

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X. Fan, Q. Zhang and D. Zhao, Eigenvalues of $p(x)$-Laplacian Dirichlet problem, J. Math. Anal. Appl., 302 (2005), 306–317. doi: 10.1016/j.jmaa.2003.11.020.  Google Scholar

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X. Fan and D. Zhao, A class of De Giorgi type and Hölder continuity, Nonlinear Anal., 36 (1999), 295–318. doi: 10.1016/S0362-546X(97)00628-7.  Google Scholar

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L. Gasiński and N. S. Papageorgiou, Anisotropic nonlinear Neumann problems, Calc. Var. Partial Differential Equations, 42 (2011), 323–354. doi: 10.1007/s00526-011-0390-2.  Google Scholar

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L. Gasiński and N. S. Papageorgiou, Multiple solutions for nonlinear coercive problems with a nonhomogeneous differential operator and a nonsmooth potential, Set-Valued Var. Anal., 20 (2012), 417–443. doi: 10.1007/s11228-011-0198-4.  Google Scholar

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L. Gasiński and N. S. Papageorgiou, Exercises in Analysis: Part 1, Problem Books in Mathematics, Springer, Cham, 2014.  Google Scholar

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J. Giacomoni, I. Schindler and P. Takáč, Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation, Ann. Scuola Norm. Sup. Pisa, Cl. Sci., 6 (2007), 117–158.  Google Scholar

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D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001.  Google Scholar

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[28]

T. C. Halsey, Electrorheological fluids, Science, 258 (1992), 761–766. Google Scholar

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show all references

References:
[1]

E. Acerbi and G. Mingione, Gradient estimates for the $p(x)$-Laplacian system, J. Reine Angew. Math., 584 (2005), 117–148. doi: 10.1515/crll.2005.2005.584.117.  Google Scholar

[2]

A. M. Alghamdi, S. Gala, C. Qian and M. A. Ragusa, The anisotropic integrability logarithmic regularity criterion for the 3D MHD equations, Electron. Res. Arch., 28 (2020), 183–193. doi: 10.3934/era.2020012.  Google Scholar

[3]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349–381. doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[4]

A. Bahrouni and V. D. Rădulescu, Singular double-phase systems with variable growth for the Baouendi-Grushin operator, Discrete Contin. Dyn. Syst., 41 (2021), 4283–4296. doi: 10.3934/dcds.2021036.  Google Scholar

[5]

A. Bahrouni, V. D. Rǎdulescu 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. Rǎdulescu 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]

J. M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. Trans. Roy. Soc. London Ser. A, 306 (1982), 557–611. doi: 10.1098/rsta.1982.0095.  Google Scholar

[8]

L. Beck and G. Mingione, Lipschitz bounds and nonuniform ellipticity, Comm. Pure Appl. Math., 73 (2020), 944–1034. doi: 10.1002/cpa.21880.  Google Scholar

[9]

D. Bonheure, P. d'Avenia and A. Pomponio, On the electrostatic Born-Infeld equation with extended charges, Comm. Math. Phys., 346 (2016), 877–906. doi: 10.1007/s00220-016-2586-y.  Google Scholar

[10]

H. Brezis and L. Nirenberg, $H^1$ versus $C^1$ local minimizers, C. R. Acad. Sci. Paris, Sér. I Math., 317 (1993), 465–472.  Google Scholar

[11]

S.-S. Byun and E. Ko, Global $C^{1, \alpha}$ regularity and existence of multiple solutions for singular $p(x)$-Laplacian equations, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 76, 29 pp. doi: 10.1007/s00526-017-1152-6.  Google Scholar

[12]

X. Chen, H. Jiang and G. Liu, Boundary spike of the singular limit of an energy minimizing problem, Discrete Contin. Dyn. Syst., 40 (2020), 3253–3290. doi: 10.3934/dcds.2020124.  Google Scholar

[13]

M. G. Crandall, P. H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations, 2 (1977), 193–222. doi: 10.1080/03605307708820029.  Google Scholar

[14]

L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Math, Vol. 2017, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.  Google Scholar

[15]

X. Fan, Q. Zhang and D. Zhao, Eigenvalues of $p(x)$-Laplacian Dirichlet problem, J. Math. Anal. Appl., 302 (2005), 306–317. doi: 10.1016/j.jmaa.2003.11.020.  Google Scholar

[16]

X. Fan and D. Zhao, A class of De Giorgi type and Hölder continuity, Nonlinear Anal., 36 (1999), 295–318. doi: 10.1016/S0362-546X(97)00628-7.  Google Scholar

[17]

N. Fukagai and K. Narukawa, On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems, Annali Mat. Pura Appl., 186 (2007), 539–564. doi: 10.1007/s10231-006-0018-x.  Google Scholar

[18]

J. P. García Azorero, I. Peral Alonso and J. J. Manfredi, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Comm. Contemp. Math., 2 (2000), 385–404. doi: 10.1142/S0219199700000190.  Google Scholar

[19]

L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis, Chapman & Hall / CRC, Boca Raton FL, 2006.  Google Scholar

[20]

L. Gasiński and N. S. Papageorgiou, Anisotropic nonlinear Neumann problems, Calc. Var. Partial Differential Equations, 42 (2011), 323–354. doi: 10.1007/s00526-011-0390-2.  Google Scholar

[21]

L. Gasiński and N. S. Papageorgiou, Multiple solutions for nonlinear coercive problems with a nonhomogeneous differential operator and a nonsmooth potential, Set-Valued Var. Anal., 20 (2012), 417–443. doi: 10.1007/s11228-011-0198-4.  Google Scholar

[22]

L. Gasiński and N. S. Papageorgiou, Exercises in Analysis: Part 1, Problem Books in Mathematics, Springer, Cham, 2014.  Google Scholar

[23]

M. Ghergu and V. Rǎdulescu, Sublinear singular elliptic problems with two parameters, J. Differential Equations, 195 (2003), 520–536. doi: 10.1016/S0022-0396(03)00105-0.  Google Scholar

[24] M. Ghergu and V. D. Rǎdulescu, Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, Clarendon Press, Oxford, 2008.   Google Scholar
[25]

J. Giacomoni, I. Schindler and P. Takáč, Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation, Ann. Scuola Norm. Sup. Pisa, Cl. Sci., 6 (2007), 117–158.  Google Scholar

[26]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001.  Google Scholar

[27]

Y. Haitao, Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem, J. Differential Equations, 189 (2003), 487–512. doi: 10.1016/S0022-0396(02)00098-0.  Google Scholar

[28]

T. C. Halsey, Electrorheological fluids, Science, 258 (1992), 761–766. Google Scholar

[29]

P. Harjuletho, P. Hästö and M. Koskenoja, Hardy's inequality in a variable exponent Sobolev space, Georgian Math. J., 12 (2005), 431–442.  Google Scholar

[30]

S. Hu and N. S. Papageorgiou, Nonlinear Neumann equations driven by a nonhomogeneous differential operator, Comm. Pure Appl. Anal., 10 (2011), 1055–1078. doi: 10.3934/cpaa.2011.10.1055.  Google Scholar

[31] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.   Google Scholar
[32]

A. C. Lazer and P. J. McKenna, On a singular nonlinear elliptic boundary value problem, Proc. Amer. Math. Soc., 111 (1991), 721–730. doi: 10.1090/S0002-9939-1991-1037213-9.  Google Scholar

[33]

G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations, Comm. Partial Differential Equations, 16 (1991), 311–361. doi: 10.1080/03605309108820761.  Google Scholar

[34]

P. Marcellini, On the definition and the lower semicontinuity of certain quasiconvex integrals, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 3 (1986), 391–409. doi: 10.1016/S0294-1449(16)30379-1.  Google Scholar

[35]

P. Marcellini, Regularity and existence of solutions of elliptic equations with $p, q$–growth conditions, J. Differential Equations, 90 (1991), 1–30. doi: 10.1016/0022-0396(91)90158-6.  Google Scholar

[36]

G. Marino and P. Winkert, Moser iteration applied to elliptic equations with critical growth on the boundary, Nonlinear Anal., 180 (2019), 154–169. doi: 10.1016/j.na.2018.10.002.  Google Scholar

[37]

G. Marino and P. Winkert, $L^\infty$-bounds for general singular elliptic equations with convection term, Appl. Math. Lett., 107 (2020), 106410, 6 pp. doi: 10.1016/j.aml.2020.106410.  Google Scholar

[38]

G. Mingione and V. Rădulescu, Recent developments in problems with nonstandard growth and nonuniform ellipticity, J. Math. Anal. Appl., 501 (2021), 125197, 41 pp. doi: 10.1016/j.jmaa.2021.125197.  Google Scholar

[39]

N. S. Papageorgiou and V. D. Rǎdulescu, Nonlinear nonhomogeneous Robin problems with a superlinear reaction term, Adv. Nonlinear Stud., 16 (2016), 737–764. doi: 10.1515/ans-2016-0023.  Google Scholar

[40]

N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, Positive solutions for nonlinear parametric singular Dirichlet problems, Bull. Math. Sci., 9 (2019), 1950011, 21 pp. doi: 10.1142/S1664360719500115.  Google Scholar

[41]

N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, Nonlinear nonhomogeneous singular problems, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 9, 31 pp. doi: 10.1007/s00526-019-1667-0.  Google Scholar

[42]

N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, Nonlinear Analysis-Theory and Methods, Springer Monographs in Mathematics, Springer Nature, Cham, 2019. doi: 10.1007/978-3-030-03430-6.  Google Scholar

[43]

N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, Anisotropic equations with indefinite potential and competing nonlinearities, Nonlinear Anal., 201 (2020), 111861, 24 pp. doi: 10.1016/j.na.2020.111861.  Google Scholar

[44]

N. S. Papageorgiou and A. Scapellato, Constant sign and nodal solutions for parametric $(p, 2)$-equations, Adv. Nonlinear Anal., 9 (2020), 449–478. doi: 10.1515/anona-2020-0009.  Google Scholar

[45]

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