Startseite Technik Variational Approaches to P(X)-Laplacian-Like Problems with Neumann Condition Originated from a Capillary Phenomena
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

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

  • Shapour Heidarkhani EMAIL logo , Ghasem A. Afrouzi und Shahin Moradi
Veröffentlicht/Copyright: 8. Februar 2018
Veröffentlichen auch Sie bei De Gruyter Brill

Abstract

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.

References

[1] Zhikov V. V., Averaging of functionals in the calculus of variations and elasticity, Math. USSR Izv. 29 (1987), 33–66.10.1070/IM1987v029n01ABEH000958Suche in Google Scholar

[2] Ružička M., Electro-rheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math., 1784, Springer, Berlin, 2000.10.1007/BFb0104030Suche in Google Scholar

[3] Chen Y., Levine S., Rao R., Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), 1383–1406.10.1137/050624522Suche in Google Scholar

[4] Antontsev S., Shmarev S., Handbook of Differential Equations, Stationary Partial Differential Equations, vol. 3, 2006 (Chapter 1).10.1016/S1874-5733(06)80005-7Suche in Google Scholar

[5] Halsey T. C., Electrorheological fluids, Science 258 (1992), 761–766.10.1126/science.258.5083.761Suche in Google Scholar PubMed

[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, 1999.10.1117/12.369269Suche in 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), 1–11.10.14232/ejqtde.2014.1.38Suche in 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.Suche in 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), 5612–5619.10.1016/j.na.2012.05.009Suche in Google Scholar

[10] Deng S. G., Positive solutions for Robin problem involving the p(x)-Laplacian, J. Math. Anal. Appl. 360 (2009), 548–560.10.1016/j.jmaa.2009.06.032Suche in Google Scholar

[11] Heidarkhani S., Ge B., Critical points approaches to elliptic problems driven by a p(x)-Laplacian, Ukrainian Math. J. 66 (2015), 1883–1903.10.1007/s11253-015-1057-5Suche in Google 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), 209–235.10.1515/anona-2014-0010Suche in Google Scholar

[13] Rădulescu V., Nonlinear elliptic equations with variable exponent: old and new, Nonlinear Anal. TMA 121 (2015), 336–369.10.1016/j.na.2014.11.007Suche in 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.Suche in Google Scholar

[15] Repovš D., Stationary waves of Schrödinger-type equations with variable exponent, Anal. Appl. 13 (2015), 645–661.10.1142/S0219530514500420Suche in 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.Suche in 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.Suche in 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), 379–399.10.1002/cpa.3160390306Suche in Google Scholar

[19] Peletier L. A., J. Serrin, Ground states for the prescribed mean curvature equation, Proc. Amer. Math. Soc. 100 (1987), 694–700.10.1090/S0002-9939-1987-0894440-8Suche in 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), 427–441.10.1515/anona-2013-0021Suche in 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), 411–426.10.1007/s00030-014-0289-7Suche in Google 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.Suche in Google Scholar

[23] Habets P., Omari P., Multiple positive solutions of a one-dimensional prescribed mean curvature problem, Commun. Contemp. Math. 9 (2007), 701–730.10.1142/S0219199707002617Suche in Google Scholar

[24] Bereanu C., Mawhin J., Boundary value problems with non-surjective ϕ-Laplacian and one-sided bounded nonlinearity, Adv. Differ. Equ. 11 (2006), 35–60.10.57262/ade/1355867723Suche in 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.Suche in Google Scholar

[26] Pan H., One-dimensional prescribed mean curvature equation with exponential nonlinearity, Nonlinear Anal. TMA 70 (2009), 999–1010.10.1016/j.na.2008.01.027Suche in 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., http://dx.doi.org/10.1080/17476933.2016.1278438.Suche in Google Scholar

[28] Avci M., Ni-Serrin type equations arising from capillarity phenomena with non-standard growth, Bound. Value Probl. 2013 (2013), 55.10.1186/1687-2770-2013-55Suche in Google Scholar

[29] Bin G., On superlinear p(x)-Laplacian-like problem without Ambrosetti and Rabinowitz condition, Bull. Korean Math. Soc. 51 (2014), 409–421.10.4134/BKMS.2014.51.2.409Suche in Google 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.Suche in Google Scholar

[31] Concus P., Finn P., A singular solution of the capillary equation I, II, Invent. Math. 29(143-148) (1975), 149–159.10.1007/BF01390191Suche in Google Scholar

[32] Heidarkhani S., Salari A., p(x)-Laplacian-like problems with Neumann condition originated from a capillary phenomena, preprintSuche in Google Scholar

[33] Manuela Rodrigues M., Multiplicity of solutions on a nonlinear eigenvalue problem for p(x)-Laplacian-like operators, Mediterr. J. Math. 9 (2012), 211–223.10.1007/s00009-011-0115-ySuche in Google Scholar

[34] Obersnel F., Omari P., Positive solutions of the Dirichlet problem for the prescribed mean curvature equation, J. Differ. Equ. 249 (2010), 1674–1725.10.1016/j.jde.2010.07.001Suche in Google 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.Suche in Google Scholar

[36] Zhou Q. M., On the superlinear problems involving p(x)-Laplacian-like operators without AR-condition, Nonlinear Anal. RWA 21 (2015), 161–169.10.1016/j.nonrwa.2014.07.003Suche in Google Scholar

[37] Chang K. C., Theory Critical Point and Applications, Shanghai Scientific and Press Technology, Shanghai, 1986.Suche in Google Scholar

[38] Willem M., Theorems Minimax, Birkhauser, Basel, (1996).10.1007/978-1-4612-4146-1Suche in Google Scholar

[39] Bonanno G., Molica Bisci G., Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Bound. Value Probl. 2009 (2009), 1–20.10.1155/2009/670675Suche in Google Scholar

[40] Ricceri B., A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000), 401–410.10.1016/S0377-0427(99)00269-1Suche in Google 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, 2016.10.1017/CBO9781316282397Suche in Google Scholar

[42] Ferrara M., Molica Bisci G., Existence results for elliptic problems with Hardy potential, Bull. Sci. Math. 138 (2014), 846–859.10.1016/j.bulsci.2014.02.002Suche in Google Scholar

[43] Galewski M., Bisci G. Molica, Existence results for one-dimensional fractional equations, Math. Meth. Appl. Sci. 39 (2016), 1480–1492.10.1002/mma.3582Suche in Google 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: 10.1007/s40840-016-0400-9.Suche in Google 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), 917–938.10.1080/10236198.2017.1306061Suche in Google 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.Suche in 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 10.1007/s00033-016-0668-5.Suche in Google 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.Suche in 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), http://dx.doi.org/10.1016/j.camwa.2016.04.012.Suche in Google 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), 493–508.10.12775/TMNA.2015.024Suche in Google Scholar

[51] Molica Bisci G., Servadei R., A bifurcation result for non-local fractional equations, Anal. Appl. 13 (2015), 371–394.10.1142/S0219530514500067Suche in Google 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), 635–660.10.57262/ade/1431115711Suche in Google Scholar

[53] Bonanno G., Candito P., Infinitely many solutions for a class of discrete non-linear boundary value problems, Appl. Anal. 88 (2009), 605–616.10.1080/00036810902942242Suche in Google Scholar

[54] Heidarkhani S., Infinitely many solutions for systems of n two-point Kirchhoff-type boundary value problems}, Ann. Polon. Math. 107 (2013), 133–152.10.4064/ap107-2-3Suche in Google Scholar

[55] Fan X. L., Zhang Q. H., Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal. TMA 52 (2003), 1843–1852.10.1016/S0362-546X(02)00150-5Suche in 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.Suche in Google Scholar

[57] Fan X. L, Zhao D., On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl. 263 (2001), 424–446.10.1006/jmaa.2000.7617Suche in Google Scholar

[58] Kováčik O., Rákosník J., On the spaces and Lp(x)(Ω) and W1,p(x)(Ω) Czechoslovak Math. 41 (1991), 592–618.10.21136/CMJ.1991.102493Suche in 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.Suche in Google Scholar

Received: 2017-5-25
Accepted: 2018-1-15
Published Online: 2018-2-8
Published in Print: 2018-4-25

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 12.1.2026 von https://www.degruyterbrill.com/document/doi/10.1515/ijnsns-2017-0114/pdf?lang=de
Button zum nach oben scrollen