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International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Armbruster, Dieter / Chen, Xi / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi

IMPACT FACTOR 2017: 1.162

CiteScore 2017: 1.41

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Mathematical Citation Quotient (MCQ) 2017: 0.12

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Volume 19, Issue 2


Variational Approaches to P(X)-Laplacian-Like Problems with Neumann Condition Originated from a Capillary Phenomena

Shapour Heidarkhani / Ghasem A. Afrouzi / Shahin Moradi
Published Online: 2018-02-08 | DOI: https://doi.org/10.1515/ijnsns-2017-0114


This article presents several sufficient conditions for the existence of at least one weak solution and infinitely many weak solutions for the following Neumann problem, originated from a capillary phenomena, {div((1+|u|p(x)1+|u|2p(x))|u|p(x)2u)+α(x)|u|p(x)2u=λf(x,u)inΩ,uν=0onΩ

where ΩRN (N2) is a bounded domain with boundary of class C1, ν is the outer unit normal to Ω, λ>0, αL(Ω), f:Ω×RR is an L1-Carathéodory function and pC0(Ω). Our technical approach is based on variational methods and we use a more precise version of Ricceri’s Variational Principle due to Bonanno and Molica Bisci. Some recent results are extended and improved. Some examples are presented to illustrate the application of our main results.

Keywords: variable exponent space; p(x)-Laplacian-likeproblem; weak solution; one solution; infinitely many solutions,variational methods

MSC 2010: Primary 35D05; Secondary 35J60


  • [1]

    Zhikov V. V., Averaging of functionals in the calculus of variations and elasticity, Math. USSR Izv. 29 (1987), 33–66.CrossrefGoogle Scholar

  • [2]

    Ružička M., Electro-rheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math., 1784, Springer, Berlin, 2000.Google Scholar

  • [3]

    Chen Y., Levine S., Rao R., Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), 1383–1406.CrossrefGoogle Scholar

  • [4]

    Antontsev S., Shmarev S., Handbook of Differential Equations, Stationary Partial Differential Equations, vol. 3, 2006 (Chapter 1).Google Scholar

  • [5]

    Halsey T. C., Electrorheological fluids, Science 258 (1992), 761–766.CrossrefGoogle Scholar

  • [6]

    Pfeiffer C., Mavroidis C., Bar-Cohen Y., Dolgin B., Electrorheological fluid based force feedback device, in: Proceedings of the 1999 SPIE Telemanipulator and Telepresence Technologies VI Conference (Boston, MA), 3840, pp. 88–99, 1999.Google Scholar

  • [7]

    Afrouzi G. A., Hadjian A., Heidarkhani S., Steklov problem involving the p(x)-Laplacian, Electronic J. Differ. Equ. Vol. 2014(134) (2014), 1–11.Google Scholar

  • [8]

    Bonanno G., Chinn&‘{i} A., Multiple solutions for elliptic problems involving the p(x)-Laplacian, Le Matematiche LXVI-Fasc. I (2011), 105–113.Google Scholar

  • [9]

    D’Aguì G., Sciammetta A., Infinitely many solutions to elliptic problems with variable exponent and nonhomogeneous Neumann conditions, Nonlinear Anal. TMA 75 (2012), 5612–5619.Google Scholar

  • [10]

    Deng S. G., Positive solutions for Robin problem involving the p(x)-Laplacian, J. Math. Anal. Appl. 360 (2009), 548–560.CrossrefGoogle Scholar

  • [11]

    Heidarkhani S., Ge B., Critical points approaches to elliptic problems driven by a p(x)-Laplacian, Ukrainian Math. J. 66 (2015), 1883–1903.CrossrefGoogle Scholar

  • [12]

    Ouaro S., Ouedraogo A., Soma S., Multivalued problem with Robin boundary condition involving diffuse measure data and variable exponent, Adv. Nonlinear Anal. 3 (2014), 209–235.Google Scholar

  • [13]

    Rădulescu V., Nonlinear elliptic equations with variable exponent: old and new, Nonlinear Anal. TMA 121 (2015), 336–369.Google Scholar

  • [14]

    Rădulescu V., Repovš D., Partial Differential Equations with Variable Exponents, Variational Methods and Qualitative Analysis, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2015.Google Scholar

  • [15]

    Repovš D., Stationary waves of Schrödinger-type equations with variable exponent, Anal. Appl. 13 (2015), 645–661.Google Scholar

  • [16]

    Ni W. M., Serrin J., Existence and non-existence theorems for ground states of quasilinear partial differential equations, The anomalous case, Atti Accd. Naz. Lincei. 77 (1986), 231–257.Google Scholar

  • [17]

    Ni W. M., Serrin J., Non-existence theorems for quasilinear partial differential equations, Rend. Circ. Mat. Palermo (2) Suppl. 8 (1985), 171–185.Google Scholar

  • [18]

    Ni W. M., Serrin J., Non-existence theorems for singular solutions of quasilinear partial differential equations, Comm. Pure Appl. Math. 39 (1986), 379–399.CrossrefGoogle Scholar

  • [19]

    Peletier L. A., J. Serrin, Ground states for the prescribed mean curvature equation, Proc. Amer. Math. Soc. 100 (1987), 694–700.Google Scholar

  • [20]

    Afrouzi G. A., Hadjian A., Molica Bisci G., Some remarks for one-dimensional mean curvature problems through a local minimization principle, Adv. Nonlinear Anal. 2 (2013), 427–441.Google Scholar

  • [21]

    Bonanno G., Livrea R., Mawhin J., Existence results for parametric boundary value problems involving the mean curvature operator, Nonlinear Differ. Equ. Appl. 22 (2015), 411–426.CrossrefGoogle Scholar

  • [22]

    Bonheure D., Habets P., Obersnel F., Omari P., Classical and non-classical positive solutions of a prescribed curvature equation with singularities, Rend. Istit. Mat. Univ. Trieste 39 (2007), 63–85.Google Scholar

  • [23]

    Habets P., Omari P., Multiple positive solutions of a one-dimensional prescribed mean curvature problem, Commun. Contemp. Math. 9 (2007), 701–730.CrossrefGoogle Scholar

  • [24]

    Bereanu C., Mawhin J., Boundary value problems with non-surjective ϕ-Laplacian and one-sided bounded nonlinearity, Adv. Differ. Equ. 11 (2006), 35–60.Google Scholar

  • [25]

    Faraci F., A note on the existence of infinitely many solutions for the one dimensional prescribed curvature equation, Stud. Univ. Babeş-Bolyai Math. 55 (2010), 83–90.Google Scholar

  • [26]

    Pan H., One-dimensional prescribed mean curvature equation with exponential nonlinearity, Nonlinear Anal. TMA 70 (2009), 999–1010.Google Scholar

  • [27]

    Afrouzi G. A., Kirane M., Shokooh S., Infinitely many weak solutions for p(x)-Laplacian-like problems with Neumann condition, Complex Var. Elliptic Equ., .CrossrefGoogle Scholar

  • [28]

    Avci M., Ni-Serrin type equations arising from capillarity phenomena with non-standard growth, Bound. Value Probl. 2013 (2013), 55.CrossrefGoogle Scholar

  • [29]

    Bin G., On superlinear p(x)-Laplacian-like problem without Ambrosetti and Rabinowitz condition, Bull. Korean Math. Soc. 51 (2014), 409–421.CrossrefGoogle Scholar

  • [30]

    Cabanillas Lapa E., Pardo Rivera V., Quique Broncano J., No-flux boundary problems involving p(x)-Laplacian-like operators, Electron. J. Diff. Equ. 2015(219) (2015), 1–10.Google Scholar

  • [31]

    Concus P., Finn P., A singular solution of the capillary equation I, II, Invent. Math. 29(143-148) (1975), 149–159.CrossrefGoogle Scholar

  • [32]

    Heidarkhani S., Salari A., p(x)-Laplacian-like problems with Neumann condition originated from a capillary phenomena, preprintGoogle Scholar

  • [33]

    Manuela Rodrigues M., Multiplicity of solutions on a nonlinear eigenvalue problem for p(x)-Laplacian-like operators, Mediterr. J. Math. 9 (2012), 211–223.Google Scholar

  • [34]

    Obersnel F., Omari P., Positive solutions of the Dirichlet problem for the prescribed mean curvature equation, J. Differ. Equ. 249 (2010), 1674–1725.CrossrefGoogle Scholar

  • [35]

    Shokooh S., Afrouzi G. A., Heidarkhani S., Multiple solutions for p(x)-Laplacian-like problems with Neumann condition, Acta Universitatis Apulensis 49 (2017), 111–128.Google Scholar

  • [36]

    Zhou Q. M., On the superlinear problems involving p(x)-Laplacian-like operators without AR-condition, Nonlinear Anal. RWA 21 (2015), 161–169.Google Scholar

  • [37]

    Chang K. C., Theory Critical Point and Applications, Shanghai Scientific and Press Technology, Shanghai, 1986.Google Scholar

  • [38]

    Willem M., Theorems Minimax, Birkhauser, Basel, (1996).Google Scholar

  • [39]

    Bonanno G., Molica Bisci G., Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Bound. Value Probl. 2009 (2009), 1–20.Google Scholar

  • [40]

    Ricceri B., A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000), 401–410.CrossrefGoogle Scholar

  • [41]

    Molica Bisci G., Rădulescu V., Servadei R., Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, vol. 162, Cambridge University Press, Cambridge, 2016.Google Scholar

  • [42]

    Ferrara M., Molica Bisci G., Existence results for elliptic problems with Hardy potential, Bull. Sci. Math. 138 (2014), 846–859.CrossrefGoogle Scholar

  • [43]

    Galewski M., Bisci G. Molica, Existence results for one-dimensional fractional equations, Math. Meth. Appl. Sci. 39 (2016), 1480–1492.CrossrefGoogle Scholar

  • [44]

    Heidarkhani S., Afrouzi G.A., Ferrara M., Caristi G., Moradi S., Existence results for impulsive damped vibration systems, Bull. Malays. Math. Sci. Soc., DOI: .CrossrefGoogle Scholar

  • [45]

    Heidarkhani S., Afrouzi G. A., Henderson J., Moradi S., Caristi G., Variational approaches to p-Laplacian discrete problems of Kirchhoff-type, J. Differ. Equ. Appl. 23 (2017), 917–938.CrossrefGoogle Scholar

  • [46]

    Heidarkhani S., G. Afrouzi A., S. Moradi, Existence of weak solutions for three-point boundary-value problems of kirchhoff-type, Electron. J. Differ. Equ. 2016(234) (2016), 1–13.Google Scholar

  • [47]

    Heidarkhani S., Afrouzi G. A., Moradi S., Caristi G., Ge B., Existence of one weak solution for p(x)-biharmonic equations with Navier boundary conditions, Zeitschrift fuer Angewandte Mathematik und Physik (2016), 67:73, DOI .CrossrefGoogle Scholar

  • [48]

    Heidarkhani S., Ferrara M., Afrouzi G. A., Caristi G., Moradi S., Existence of solutions for Dirichlet quasilinear systems including a nonlinear function of the derivative, Electronic J. Diff. Equ., Vol. 2016(56) (2016), 1–12.Google Scholar

  • [49]

    Heidarkhani S., Zhou Y., Caristi G., Afrouzi G. A., Moradi S., Existence results for fractional differential systems through a local minimization principle, Comput. Math. Appl. (2016), .CrossrefGoogle Scholar

  • [50]

    Molica Bisci G., Rădulescu V., Bifurcation analysis of a singular elliptic problem modelling the equilibrium of anisotropic continuous media, Topol. Methods Nonlinear Anal. 45 (2015), 493–508.CrossrefGoogle Scholar

  • [51]

    Molica Bisci G., Servadei R., A bifurcation result for non-local fractional equations, Anal. Appl. 13 (2015), 371–394.CrossrefGoogle Scholar

  • [52]

    Molica Bisci G., Servadei R., Lower semicontinuity of functionals of fractional type and applications to nonlocal equations with critical Sobolev exponent, Adv. Differ. Equ. 20 (2015), 635–660.Google Scholar

  • [53]

    Bonanno G., Candito P., Infinitely many solutions for a class of discrete non-linear boundary value problems, Appl. Anal. 88 (2009), 605–616.CrossrefGoogle Scholar

  • [54]

    Heidarkhani S., Infinitely many solutions for systems of n two-point Kirchhoff-type boundary value problems}, Ann. Polon. Math. 107 (2013), 133–152.Crossref

  • [55]

    Fan X. L., Zhang Q. H., Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal. TMA 52 (2003), 1843–1852.Google Scholar

  • [56]

    Fan X. L., Zhao D., On the generalize Orlicz-Sobolev space Wk,p(x)(Ω), J. Gansu Educ. College 12 (1998), 1–6.

  • [57]

    Fan X. L, Zhao D., On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl. 263 (2001), 424–446.Crossref

  • [58]

    Kováčik O., Rákosník J., On the spaces and Lp(x)(Ω) and W1,p(x)(Ω) Czechoslovak Math. 41 (1991), 592–618.Google Scholar

  • [59]

    Sanko S. G., Denseness of C0(ℝN) in the generalized Sobolev spaces Wm,p(x)(ℝN), Dokl. Ross. Akad. Nauk. 369 (1999), 451–454.Google Scholar

About the article

Received: 2017-05-25

Accepted: 2018-01-15

Published Online: 2018-02-08

Published in Print: 2018-04-25

Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 19, Issue 2, Pages 189–203, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2017-0114.

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