doi: 10.3934/dcdss.2022135
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Singular equations with variable exponents and concave-convex nonlinearities

1. 

Department of Mathematics, Pedagogical University of Cracow, Podchorazych 2, 30-084 Cracow, Poland

2. 

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

Corresponding author: Leszek Gasiński

Received  February 2022 Revised  May 2022 Early access July 2022

We consider a nonlinear anisotropic Dirichlet problem with a reaction that has the combined effects of three distinct nonlinearities: a parametric singular term, a parametric "concave" term (the parameter $ \lambda>0 $ is the same in both) and a nonparametric "convex" perturbation. So, the problem is a singular version of the well known "concave-convex" problem. We prove an existence and multiplicity result which is global in the parameter $ \lambda>0 $ (a bifurcation-type theorem). We also indicate some small improvements in the case of $ (p(z),q(z)) $ and $ p(z) $-equations.

Citation: Leszek Gasiński, Nikolaos S. Papageorgiou. Singular equations with variable exponents and concave-convex nonlinearities. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022135
References:
[1]

A. AmbrosettiH. Brézis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078.

[2]

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. doi: 10.1007/s00526-017-1152-6.

[3]

D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces, Foundations and Harmonic Analysis, Birkhäuser/Springer, Heidelberg, 2013. doi: 10.1007/978-3-0348-0548-3.

[4]

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

[5]

X. Fan, Global $C^{1, \alpha}$ regularity for variable exponent elliptic equations in divergence form, J. Differential Equations, 235 (2007), 397-417.  doi: 10.1016/j.jde.2007.01.008.

[6]

X. FanQ. 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.

[7]

J. P. García AzoreroI. Peral Alonso and J. J. Manfredi, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Commun. Contemp. Math., 2 (2000), 385-404.  doi: 10.1142/S0219199700000190.

[8]

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

[9]

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.

[10]

L. Gasiński and N. S. Papageorgiou, Exercises in Analysis. Part 2. Nonlinear Analysis, Springer, Cham, 2016. doi: 10.1007/978-3-319-27817-9.

[11]

J. GiacomoniI. Schindler and P. Takáč, Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 6 (2007), 117-158. 

[12]

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

[13]

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

[14]

E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer-Verlag, New York-Heidelberg, 1975.

[15]

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.

[16]

N. S. Papageorgiou, V. D. R$ {\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over a} }} $dulescu 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.

[17]

N. S. Papageorgiou, V. D. R$ {\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over a} }} $dulescu 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.

[18]

N. S. Papageorgiou, V. D. R$ {\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over a} }} $dulescu and D. D. Repovš, Nonlinear Analysis – Theory and Methods, Springer, Cham, 2019. doi: 10.1007/978-3-030-03430-6.

[19]

N. S. PapageorgiouV. D. R$ {\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over a} }} $dulescu and Y. Zhang, Anisotropic singular double phase Dirichlet problems, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 4465-4502.  doi: 10.3934/dcdss.2021111.

[20]

N. S. Papageorgiou and P. Winkert, $(p, q)$-equations with singular and concave convex nonlinearities, Appl. Math. Optim., 84 (2021), 2601-2628.  doi: 10.1007/s00245-020-09720-0.

[21]

N. S. Papageorgiou and P. Winkert, Positive solutions for singular anisotropic $(p, q)$-equations, J. Geom. Anal., 31 (2021), 11849-11877.  doi: 10.1007/s12220-021-00703-3.

[22]

K. Saoudi and A. Ghanmi, A multiplicity results for a singular equation involving the $p(x)$-Laplace operator, Complex Var. Elliptic Equ., 62 (2017), 695-725.  doi: 10.1080/17476933.2016.1238466.

[23]

M. Zhang, The rotation number approach to eigenvalues of the one-dimensional $p$-Laplacian with periodic potentials, J. London Math. Soc. (2), 64 (2001), 125-143.  doi: 10.1017/S0024610701002277.

show all references

References:
[1]

A. AmbrosettiH. Brézis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078.

[2]

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. doi: 10.1007/s00526-017-1152-6.

[3]

D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces, Foundations and Harmonic Analysis, Birkhäuser/Springer, Heidelberg, 2013. doi: 10.1007/978-3-0348-0548-3.

[4]

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

[5]

X. Fan, Global $C^{1, \alpha}$ regularity for variable exponent elliptic equations in divergence form, J. Differential Equations, 235 (2007), 397-417.  doi: 10.1016/j.jde.2007.01.008.

[6]

X. FanQ. 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.

[7]

J. P. García AzoreroI. Peral Alonso and J. J. Manfredi, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Commun. Contemp. Math., 2 (2000), 385-404.  doi: 10.1142/S0219199700000190.

[8]

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

[9]

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.

[10]

L. Gasiński and N. S. Papageorgiou, Exercises in Analysis. Part 2. Nonlinear Analysis, Springer, Cham, 2016. doi: 10.1007/978-3-319-27817-9.

[11]

J. GiacomoniI. Schindler and P. Takáč, Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 6 (2007), 117-158. 

[12]

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

[13]

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

[14]

E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer-Verlag, New York-Heidelberg, 1975.

[15]

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.

[16]

N. S. Papageorgiou, V. D. R$ {\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over a} }} $dulescu 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.

[17]

N. S. Papageorgiou, V. D. R$ {\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over a} }} $dulescu 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.

[18]

N. S. Papageorgiou, V. D. R$ {\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over a} }} $dulescu and D. D. Repovš, Nonlinear Analysis – Theory and Methods, Springer, Cham, 2019. doi: 10.1007/978-3-030-03430-6.

[19]

N. S. PapageorgiouV. D. R$ {\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over a} }} $dulescu and Y. Zhang, Anisotropic singular double phase Dirichlet problems, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 4465-4502.  doi: 10.3934/dcdss.2021111.

[20]

N. S. Papageorgiou and P. Winkert, $(p, q)$-equations with singular and concave convex nonlinearities, Appl. Math. Optim., 84 (2021), 2601-2628.  doi: 10.1007/s00245-020-09720-0.

[21]

N. S. Papageorgiou and P. Winkert, Positive solutions for singular anisotropic $(p, q)$-equations, J. Geom. Anal., 31 (2021), 11849-11877.  doi: 10.1007/s12220-021-00703-3.

[22]

K. Saoudi and A. Ghanmi, A multiplicity results for a singular equation involving the $p(x)$-Laplace operator, Complex Var. Elliptic Equ., 62 (2017), 695-725.  doi: 10.1080/17476933.2016.1238466.

[23]

M. Zhang, The rotation number approach to eigenvalues of the one-dimensional $p$-Laplacian with periodic potentials, J. London Math. Soc. (2), 64 (2001), 125-143.  doi: 10.1017/S0024610701002277.

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