April  2019, 12(2): 139-150. doi: 10.3934/dcdss.2019010

Perturbation effects for the minimal surface equation with multiple variable exponents

Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

* Corresponding author

This paper is dedicated to Prof. Vicenţiu Rǎdulescu on the occasion of his 60th anniversary

Received  May 2017 Revised  November 2017 Published  August 2018

We are concerned with the existence of nontrivial weak solutions for a class of generalized minimal surface equations with subcritical growth and Dirichlet boundary condition. In relationship with the values of several variable exponents, we establish two sufficient conditions for the existence of solutions. In the first part of this paper, we prove the existence of a non-negative solution. Next, we are concerned with the existence of infinitely many solutions in a symmetric abstract setting.

Citation: Ramzi Alsaedi. Perturbation effects for the minimal surface equation with multiple variable exponents. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 139-150. doi: 10.3934/dcdss.2019010
References:
[1]

R. Alsaedi, Perturbed subcritical Dirichlet problems with variable exponents, Electron. J. Differential Equations, 2016 (2016), Paper No. 295, 12 pp.  Google Scholar

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R. AlsaediH. Mȃagli and N. Zeddini, Exact behavior of the unique positive solution to some singular elliptic problem in exterior domains, Nonlinear Anal., 119 (2015), 186-198.  doi: 10.1016/j.na.2014.09.018.  Google Scholar

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

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L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal., 19 (1992), 581-597.  doi: 10.1016/0362-546X(92)90023-8.  Google Scholar

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H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal., 10 (1986), 55-64.  doi: 10.1016/0362-546X(86)90011-8.  Google Scholar

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

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

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D. Edmunds, J. Lang and O. Méndez, Differential Operators on Spaces of Variable Integrability, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014. doi: 10.1142/9124.  Google Scholar

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I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[10]

X. Fan and Q. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal., 52 (2003), 1843-1852.  doi: 10.1016/S0362-546X(02)00150-5.  Google Scholar

[11]

Y. Fu and Y. Shan, On the removability of isolated singular points for elliptic equations involving variable exponent, Adv. Nonlinear Anal., 5 (2016), 121-132.  doi: 10.1515/anona-2015-0055.  Google Scholar

[12]

E. Giusti, On the equation of surfaces of prescribed mean curvature: existence and uniqueness without boundary conditions, Invent. Math., 46 (1978), 111-137.  doi: 10.1007/BF01393250.  Google Scholar

[13]

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

[14]

B. Kawohl, From p-Laplace to mean curvature operator and related questions, Progress in Partial Differential Equations: The Metz Surveys, 40-56, Pitman Res. Notes Math. Ser., 249, Longman Sci. Tech., Harlow, 1991.  Google Scholar

[15]

I. Kim and Y. Kim, 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.  Google Scholar

[16]

H. Maagli, R. Alsaedi and N. Zeddini, Bifurcation analysis of elliptic equations described by nonhomogeneous differential operators, Electron. J. Differential Equations, 2017 (2017), Paper No. 223, 12 pp.  Google Scholar

[17]

M. Mihǎilescu and V. Rǎdulescu, On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc., 135 (2007), 2929-2937.  doi: 10.1090/S0002-9939-07-08815-6.  Google Scholar

[18]

P. Pucci and V. Rǎdulescu, The impact of the mountain pass theory in nonlinear analysis: A mathematical survey, Boll. Unione Mat. Ital., Series Ⅸ, 3 (2010), 543-582.  Google Scholar

[19]

P. Pucci and Q. 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.  Google Scholar

[20]

V. Rǎdulescu, Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations. Monotonicity, Analytic, and Variational Methods, Contemporary Mathematics and Its Applications, vol. 6, Hindawi Publishing Corporation, New York, 2008. doi: 10.1155/9789774540394.  Google Scholar

[21]

V. Rǎdulescu, 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.  Google Scholar

[22]

V. Rǎdulescu and D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, CRC Press, Taylor & Francis Group, Boca Raton FL, 2015. doi: 10.1201/b18601.  Google Scholar

[23]

M. Struwe, Plateau's Problem and the Calculus of Variations, Mathematical Notes, vol. 35, Princeton University Press, Princeton, NJ, 1988.  Google Scholar

show all references

References:
[1]

R. Alsaedi, Perturbed subcritical Dirichlet problems with variable exponents, Electron. J. Differential Equations, 2016 (2016), Paper No. 295, 12 pp.  Google Scholar

[2]

R. AlsaediH. Mȃagli and N. Zeddini, Exact behavior of the unique positive solution to some singular elliptic problem in exterior domains, Nonlinear Anal., 119 (2015), 186-198.  doi: 10.1016/j.na.2014.09.018.  Google Scholar

[3]

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

[4]

L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal., 19 (1992), 581-597.  doi: 10.1016/0362-546X(92)90023-8.  Google Scholar

[5]

H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal., 10 (1986), 55-64.  doi: 10.1016/0362-546X(86)90011-8.  Google Scholar

[6]

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

[7]

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

[8]

D. Edmunds, J. Lang and O. Méndez, Differential Operators on Spaces of Variable Integrability, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014. doi: 10.1142/9124.  Google Scholar

[9]

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[10]

X. Fan and Q. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal., 52 (2003), 1843-1852.  doi: 10.1016/S0362-546X(02)00150-5.  Google Scholar

[11]

Y. Fu and Y. Shan, On the removability of isolated singular points for elliptic equations involving variable exponent, Adv. Nonlinear Anal., 5 (2016), 121-132.  doi: 10.1515/anona-2015-0055.  Google Scholar

[12]

E. Giusti, On the equation of surfaces of prescribed mean curvature: existence and uniqueness without boundary conditions, Invent. Math., 46 (1978), 111-137.  doi: 10.1007/BF01393250.  Google Scholar

[13]

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

[14]

B. Kawohl, From p-Laplace to mean curvature operator and related questions, Progress in Partial Differential Equations: The Metz Surveys, 40-56, Pitman Res. Notes Math. Ser., 249, Longman Sci. Tech., Harlow, 1991.  Google Scholar

[15]

I. Kim and Y. Kim, 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.  Google Scholar

[16]

H. Maagli, R. Alsaedi and N. Zeddini, Bifurcation analysis of elliptic equations described by nonhomogeneous differential operators, Electron. J. Differential Equations, 2017 (2017), Paper No. 223, 12 pp.  Google Scholar

[17]

M. Mihǎilescu and V. Rǎdulescu, On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc., 135 (2007), 2929-2937.  doi: 10.1090/S0002-9939-07-08815-6.  Google Scholar

[18]

P. Pucci and V. Rǎdulescu, The impact of the mountain pass theory in nonlinear analysis: A mathematical survey, Boll. Unione Mat. Ital., Series Ⅸ, 3 (2010), 543-582.  Google Scholar

[19]

P. Pucci and Q. 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.  Google Scholar

[20]

V. Rǎdulescu, Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations. Monotonicity, Analytic, and Variational Methods, Contemporary Mathematics and Its Applications, vol. 6, Hindawi Publishing Corporation, New York, 2008. doi: 10.1155/9789774540394.  Google Scholar

[21]

V. Rǎdulescu, 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.  Google Scholar

[22]

V. Rǎdulescu and D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, CRC Press, Taylor & Francis Group, Boca Raton FL, 2015. doi: 10.1201/b18601.  Google Scholar

[23]

M. Struwe, Plateau's Problem and the Calculus of Variations, Mathematical Notes, vol. 35, Princeton University Press, Princeton, NJ, 1988.  Google Scholar

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