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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

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