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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. [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. [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. [53] M. A. Ragusa and A. Tachikawa, Regularity for minimizers for functionals of double phase with variable exponents, Adv. Nonlinear Anal., 9 (2020), 710–728. doi: 10.1515/anona-2020-0022. [54] K. Saoudi and A. Ghanmi, A multiplicity result for a singular equation involving the $p(x)$-Laplace operator, Complex Var. Elliptic Equ., 62 (2017), 695–725. doi: 10.1080/17476933.2016.1238466. [55] Y. Sun, S. Wu and Y. Long, Combined effects of singular and superlinear nonlinearities in some singular boundary value problems, J. Differential Equations, 176 (2001), 511–531. doi: 10.1006/jdeq.2000.3973. [56] P. Winkert, $L^\infty$-estimates for nonlinear elliptic Neumann boundary value problems, Nonlin. Differ. Equ. Appl. (NoDEA), 17 (2010), 289–310. doi: 10.1007/s00030-009-0054-5. [57] W. M. Winslow, Induced fibration of suspensions, J. Appl. Phys., 20 (1949), 1137–1140. doi: 10.1063/1.1698285. [58] M. Xiang, B. Zhang and D. Hu, Kirchhoff-type differential inclusion problems involving the fractional Laplacian and strong damping, Electron. Res. Arch., 28 (2020), 651–669. doi: 10.3934/era.2020034. [59] Q. Zhang, A strong maximum principle for differential equations with nonstandard $p(x)$-growth conditions, J. Math. Anal. Appl., 312 (2005), 24–32. doi: 10.1016/j.jmaa.2005.03.013. [60] Q. Zhang and V. D. Rǎdulescu, 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. [61] Y. Zhang and M. Feng, A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior, Electron. Res. Arch., 28 (2020), 1419–1438. doi: 10.3934/era.2020075. [62] M. Zhen, B. Zhang and V. D. Rădulescu, Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case, Discrete Contin. Dyn. Syst., 41 (2021), 2653–2676. doi: 10.3934/dcds.2020379. [63] V. V. Zhikov, On variational problems and nonlinear elliptic equations with nonstandard growth conditions, J. Math. Sci., 173 (2011), 463–570. doi: 10.1007/s10958-011-0260-7.

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. 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