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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 and Continuous Dynamical Systems - S, 2021, 14 (12) : 4465-4502. doi: 10.3934/dcdss.2021111
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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

[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.

[22]

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

[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.

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

[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.

[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.

[28]

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

[29]

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

[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.

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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.

[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.

[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.

[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.

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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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[45]

N. S. Papageorgiou, C. Vetro and F. Vetro, Parametric nonlinear singular Dirichlet problems, Nonlinear Anal. Real World Appl., 45 (2019), 239–254. doi: 10.1016/j.nonrwa.2018.07.006.

[46]

N. S. Papageorgiou, C. Vetro and F. Vetro, Multiple solutions for $(p, 2)$-equations at resonance, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 347–374. doi: 10.3934/dcdss.2019024.

[47]

N. S. Papageorgiou, C. Vetro and F. Vetro, Multiple solutions for parametric double phase Dirichlet problems, Comm. Contemp. Math., 23 (2021), 2050006, 18 pp. doi: 10.1142/S0219199720500066.

[48]

N. S. Papageorgiou and P. Winkert, Applied Nonlinear Functional Analysis, De Gruyter, Berlin, 2018.

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N. S. Papageorgiou and P. Winkert, Singular $p$-Laplacian equations with superlinear perturbation, J. Differential Equations, 266 (2019), 1462–1487. doi: 10.1016/j.jde.2018.08.002.

[50]

N. S. Papageorgiou and C. Zhang, Noncoercive resonant $(p, 2)$-equations with concave terms, Adv. Nonlinear Anal., 9 (2020), 228–249. doi: 10.1515/anona-2018-0175.

[51]

N. S. Papageorgiou and Y. Zhang, Constant sign and nodal solutions for superlinear $(p, q)$-equations with indefinite potential and concave boundary condition, Adv. Nonlinear Anal., 10 (2021), 76–101. doi: 10.1515/anona-2020-0101.

[52] V. D. Rǎdulescu and D. D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, CRC Press, Boca Raton, FL, 2015.  doi: 10.1201/b18601.
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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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[22]

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

[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.

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

[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.

[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.

[28]

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

[29]

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

[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.

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[45]

N. S. Papageorgiou, C. Vetro and F. Vetro, Parametric nonlinear singular Dirichlet problems, Nonlinear Anal. Real World Appl., 45 (2019), 239–254. doi: 10.1016/j.nonrwa.2018.07.006.

[46]

N. S. Papageorgiou, C. Vetro and F. Vetro, Multiple solutions for $(p, 2)$-equations at resonance, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 347–374. doi: 10.3934/dcdss.2019024.

[47]

N. S. Papageorgiou, C. Vetro and F. Vetro, Multiple solutions for parametric double phase Dirichlet problems, Comm. Contemp. Math., 23 (2021), 2050006, 18 pp. doi: 10.1142/S0219199720500066.

[48]

N. S. Papageorgiou and P. Winkert, Applied Nonlinear Functional Analysis, De Gruyter, Berlin, 2018.

[49]

N. S. Papageorgiou and P. Winkert, Singular $p$-Laplacian equations with superlinear perturbation, J. Differential Equations, 266 (2019), 1462–1487. doi: 10.1016/j.jde.2018.08.002.

[50]

N. S. Papageorgiou and C. Zhang, Noncoercive resonant $(p, 2)$-equations with concave terms, Adv. Nonlinear Anal., 9 (2020), 228–249. doi: 10.1515/anona-2018-0175.

[51]

N. S. Papageorgiou and Y. Zhang, Constant sign and nodal solutions for superlinear $(p, q)$-equations with indefinite potential and concave boundary condition, Adv. Nonlinear Anal., 10 (2021), 76–101. doi: 10.1515/anona-2020-0101.

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