July  2020, 13(7): 2057-2068. doi: 10.3934/dcdss.2020158

Combined effects for non-autonomous singular biharmonic problems

a. 

Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia

b. 

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

c. 

Institute of Mathematics "Simion Stoilow" of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania

d. 

Faculty of Education and Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia

* Corresponding author: Vicenţiu D. Rădulescu

To Professor Patrizia Pucci, on the occasion of her 65th birthday. Her work and friendship are a permanent source of inspiration and motivation.

Received  July 2018 Revised  August 2018 Published  November 2019

We study the existence of nontrivial weak solutions for a class of generalized $ p(x) $-biharmonic equations with singular nonlinearity and Navier boundary condition. The proofs combine variational and topological arguments. The approach developed in this paper allows for the treatment of several classes of singular biharmonic problems with variable growth arising in applied sciences, including the capillarity equation and the mean curvature problem.

Citation: Vicenţiu D. Rădulescu, Dušan D. Repovš. Combined effects for non-autonomous singular biharmonic problems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 2057-2068. doi: 10.3934/dcdss.2020158
References:
[1]

G. Autuori, F. Colasuonno and P. Pucci, On the existence of stationary solutions for higher-order p-Kirchhoff problems, Commun. Contemp. Math., 16 (2014), 1450002, 43 pp. doi: 10.1142/S0219199714500023.

[2]

G. Autuori and P. Pucci, Asymptotic stability for Kirchhoff systems in variable exponent Sobolev spaces, Complex Var. Elliptic Equ., 56 (2011), 715-753.  doi: 10.1080/17476931003786691.

[3]

A. Ayoujil and A. El Amrouss, Continuous spectrum of a fourth-order nonhomogeneous elliptic equation with variable exponent, Electron. J. Differ. Equations, 2011 (2011), 12 pp.

[4]

P. BaroniM. Colombo and G. Mingione, Non-autonomous functionals, borderline cases and related function classes, Algebra i Analiz, 27 (2015), 6-50.  doi: 10.1090/spmj/1392.

[5]

H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Collection Mathématiques Appliquées pour la Maâtrise, Masson, Paris, 1983.

[6]

M. Cencelj, V. D. Rădulescu and D. D. Repovš, Double phase problems with variable growth, Nonlinear Anal., 177 (2018), part A, 270–287. doi: 10.1016/j.na.2018.03.016.

[7]

M. CenceljD. Repovš and Ž. Virk, Multiple perturbations of a singular eigenvalue problem, Nonlinear Anal., 119 (2015), 37-45.  doi: 10.1016/j.na.2014.07.015.

[8]

Y. M. ChenS. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406.  doi: 10.1137/050624522.

[9]

M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Archive for Rational Mechanics and Analysis, 218 (2015), 219-273.  doi: 10.1007/s00205-015-0859-9.

[10]

M. Colombo and G. Mingione, Calderón-Zygmund estimates and non-uniformly elliptic operators, Journal of Functional Analysis, 270 (2016), 1416-1478.  doi: 10.1016/j.jfa.2015.06.022.

[11]

L. Diening, P. Hästö, P. Harjulehto and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Springer Lecture Notes, vol. 2017, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-18363-8.

[12]

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

[13]

P. HarjulehtoP. HästöÚ. V. Le and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal., 72 (2010), 4551-4574.  doi: 10.1016/j.na.2010.02.033.

[14]

K. Kefi, V. D. Rădulescu, On a p(x)-biharmonic problem with singular weights, Z. Angew. Math. Phys., 68 (2017), Art. 80, 13 pp. doi: 10.1007/s00033-017-0827-3.

[15]

K. Kefi and K. Saoudi, On the existence of a weak solution for some singular p(x)-biharmonic equation with Navier boundary conditions, Advances in Nonlinear Analysis, 8 (2019), 1171-1183.  doi: 10.1515/anona-2016-0260.

[16]

I. H. Kim and Y.-H. Kim nd, Mountain pass type solutions and positivity of the infimum eigenvalue for quasilinear elliptic equations with variable exponents, Manuscripta Math., 147 (2015), 169-191.  doi: 10.1007/s00229-014-0718-2.

[17]

J. J. LiuP. PucciH. T. Wu and Q. H. Zhang, Existence and blow-up rate of large solutions of p(x)-Laplacian equations with gradient terms, J. Math. Anal. Appl., 457 (2018), 944-977.  doi: 10.1016/j.jmaa.2017.08.038.

[18]

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.

[19]

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.

[20]

M. Mihăilescu and V. Rădulescu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 2625-2641.  doi: 10.1098/rspa.2005.1633.

[21]

J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer-Verlag, Berlin, 1983. doi: 10.1007/BFb0072210.

[22]

W. Orlicz, Über konjugierte Exponentenfolgen, Studia Math., 3 (1931), 200-211.  doi: 10.4064/sm-3-1-200-211.

[23]

P. Pucci and Q. H. Zhang, Existence of entire solutions for a class of variable exponent elliptic equations, J. Differential Equations, 257 (2014), 1529-1566.  doi: 10.1016/j.jde.2014.05.023.

[24]

V. D. Rădulescu, Nonlinear elliptic equations with variable exponent: Old and new, Nonlinear Analysis: Theory, Methods and Applications, 121 (2015), 336-369.  doi: 10.1016/j.na.2014.11.007.

[25]

V. D. Rădulescu and D. D. Repovš, Combined effects in nonlinear problems arising in the study of anisotropic continuous media, Nonlinear Anal., 75 (2012), 1524-1530.  doi: 10.1016/j.na.2011.01.037.

[26] V. D. Rădulescu and D. D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2015.  doi: 10.1201/b18601.
[27]

V. D. Rădulescu and Q. H. Zhang, Double phase anisotropic variational problems and combined effects of reaction and absorption terms, J. Math. Pures Appl. (9), 118 (2018), 159–203. doi: 10.1016/j.matpur.2018.06.015.

[28]

D. D. Repovš, Stationary waves of Schrödinger-type equations with variable exponent, Anal. Appl. (Singap.), 13 (2015), 645-661.  doi: 10.1142/S0219530514500420.

[29]

M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, 1748. Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.

[30]

X. Y. Shi, V. D. Rădulescu, D. D. Repovš and Q. H. Zhang, Multiple solutions of double phase variational problems with variable exponent, Advances in Calculus of Variations, (2018), https://doi.org/10.1515/acv-2018-0003.

[31]

J. Simon, Régularité de la solution d'une équation non linéaire dans $ \mathbb R^N$, Journées d'Analyse Non Linéaire (Proc. Conf., Besançon, 1977), Lecture Notes in Math., Springer, Berlin, 665 (1978), 205–227.

[32]

A. B. Zang and Y. Fu, Interpolation inequalities for derivatives in variable exponent Lebesgue Sobolev spaces, Nonlinear Anal., 69 (2008), 3629-3636.  doi: 10.1016/j.na.2007.10.001.

[33]

V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675–710,877.

[34]

V. V. Zhikov, Lavrentiev phenomenon and homogenization for some variational problems, C. R. Acad. Sci. Paris Sér. I Math., 316 (1993), 435-439. 

show all references

To Professor Patrizia Pucci, on the occasion of her 65th birthday. Her work and friendship are a permanent source of inspiration and motivation.

References:
[1]

G. Autuori, F. Colasuonno and P. Pucci, On the existence of stationary solutions for higher-order p-Kirchhoff problems, Commun. Contemp. Math., 16 (2014), 1450002, 43 pp. doi: 10.1142/S0219199714500023.

[2]

G. Autuori and P. Pucci, Asymptotic stability for Kirchhoff systems in variable exponent Sobolev spaces, Complex Var. Elliptic Equ., 56 (2011), 715-753.  doi: 10.1080/17476931003786691.

[3]

A. Ayoujil and A. El Amrouss, Continuous spectrum of a fourth-order nonhomogeneous elliptic equation with variable exponent, Electron. J. Differ. Equations, 2011 (2011), 12 pp.

[4]

P. BaroniM. Colombo and G. Mingione, Non-autonomous functionals, borderline cases and related function classes, Algebra i Analiz, 27 (2015), 6-50.  doi: 10.1090/spmj/1392.

[5]

H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Collection Mathématiques Appliquées pour la Maâtrise, Masson, Paris, 1983.

[6]

M. Cencelj, V. D. Rădulescu and D. D. Repovš, Double phase problems with variable growth, Nonlinear Anal., 177 (2018), part A, 270–287. doi: 10.1016/j.na.2018.03.016.

[7]

M. CenceljD. Repovš and Ž. Virk, Multiple perturbations of a singular eigenvalue problem, Nonlinear Anal., 119 (2015), 37-45.  doi: 10.1016/j.na.2014.07.015.

[8]

Y. M. ChenS. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406.  doi: 10.1137/050624522.

[9]

M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Archive for Rational Mechanics and Analysis, 218 (2015), 219-273.  doi: 10.1007/s00205-015-0859-9.

[10]

M. Colombo and G. Mingione, Calderón-Zygmund estimates and non-uniformly elliptic operators, Journal of Functional Analysis, 270 (2016), 1416-1478.  doi: 10.1016/j.jfa.2015.06.022.

[11]

L. Diening, P. Hästö, P. Harjulehto and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Springer Lecture Notes, vol. 2017, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-18363-8.

[12]

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

[13]

P. HarjulehtoP. HästöÚ. V. Le and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal., 72 (2010), 4551-4574.  doi: 10.1016/j.na.2010.02.033.

[14]

K. Kefi, V. D. Rădulescu, On a p(x)-biharmonic problem with singular weights, Z. Angew. Math. Phys., 68 (2017), Art. 80, 13 pp. doi: 10.1007/s00033-017-0827-3.

[15]

K. Kefi and K. Saoudi, On the existence of a weak solution for some singular p(x)-biharmonic equation with Navier boundary conditions, Advances in Nonlinear Analysis, 8 (2019), 1171-1183.  doi: 10.1515/anona-2016-0260.

[16]

I. H. Kim and Y.-H. Kim nd, Mountain pass type solutions and positivity of the infimum eigenvalue for quasilinear elliptic equations with variable exponents, Manuscripta Math., 147 (2015), 169-191.  doi: 10.1007/s00229-014-0718-2.

[17]

J. J. LiuP. PucciH. T. Wu and Q. H. Zhang, Existence and blow-up rate of large solutions of p(x)-Laplacian equations with gradient terms, J. Math. Anal. Appl., 457 (2018), 944-977.  doi: 10.1016/j.jmaa.2017.08.038.

[18]

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.

[19]

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.

[20]

M. Mihăilescu and V. Rădulescu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 2625-2641.  doi: 10.1098/rspa.2005.1633.

[21]

J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer-Verlag, Berlin, 1983. doi: 10.1007/BFb0072210.

[22]

W. Orlicz, Über konjugierte Exponentenfolgen, Studia Math., 3 (1931), 200-211.  doi: 10.4064/sm-3-1-200-211.

[23]

P. Pucci and Q. H. Zhang, Existence of entire solutions for a class of variable exponent elliptic equations, J. Differential Equations, 257 (2014), 1529-1566.  doi: 10.1016/j.jde.2014.05.023.

[24]

V. D. Rădulescu, Nonlinear elliptic equations with variable exponent: Old and new, Nonlinear Analysis: Theory, Methods and Applications, 121 (2015), 336-369.  doi: 10.1016/j.na.2014.11.007.

[25]

V. D. Rădulescu and D. D. Repovš, Combined effects in nonlinear problems arising in the study of anisotropic continuous media, Nonlinear Anal., 75 (2012), 1524-1530.  doi: 10.1016/j.na.2011.01.037.

[26] V. D. Rădulescu and D. D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2015.  doi: 10.1201/b18601.
[27]

V. D. Rădulescu and Q. H. Zhang, Double phase anisotropic variational problems and combined effects of reaction and absorption terms, J. Math. Pures Appl. (9), 118 (2018), 159–203. doi: 10.1016/j.matpur.2018.06.015.

[28]

D. D. Repovš, Stationary waves of Schrödinger-type equations with variable exponent, Anal. Appl. (Singap.), 13 (2015), 645-661.  doi: 10.1142/S0219530514500420.

[29]

M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, 1748. Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.

[30]

X. Y. Shi, V. D. Rădulescu, D. D. Repovš and Q. H. Zhang, Multiple solutions of double phase variational problems with variable exponent, Advances in Calculus of Variations, (2018), https://doi.org/10.1515/acv-2018-0003.

[31]

J. Simon, Régularité de la solution d'une équation non linéaire dans $ \mathbb R^N$, Journées d'Analyse Non Linéaire (Proc. Conf., Besançon, 1977), Lecture Notes in Math., Springer, Berlin, 665 (1978), 205–227.

[32]

A. B. Zang and Y. Fu, Interpolation inequalities for derivatives in variable exponent Lebesgue Sobolev spaces, Nonlinear Anal., 69 (2008), 3629-3636.  doi: 10.1016/j.na.2007.10.001.

[33]

V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675–710,877.

[34]

V. V. Zhikov, Lavrentiev phenomenon and homogenization for some variational problems, C. R. Acad. Sci. Paris Sér. I Math., 316 (1993), 435-439. 

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