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Existence results for a non-homogeneous Neumann problem through Orlicz–Sobolev spaces

  • Shapour Heidarkhani EMAIL logo , Giuseppe Caristi , Ghasem A. Afrouzi and Shahin Moradi
Published/Copyright: November 27, 2019
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Abstract

Based on a variational principle for smooth functionals defined on reflexive Banach spaces, the existence of at least one weak solution for a non-homogeneous Neumann problem in an appropriate Orlicz–Sobolev space is discussed.

MSC 2010: 35D05; 35J60; 46N20

Acknowledgements

We would like to show our great thanks to Professor Yong Zhou for his valuable suggestions and comments, which improved the former version of this paper and made us rewrite the paper in a more clear way.

References

[1] R. A. Adams, Sobolev Spaces, Pure Appl. Math. 65, Academic Press, New York, 1975. Search in Google Scholar

[2] G. A. Afrouzi, A. Hadjian and G. Molica Bisci, Some remarks for one-dimensional mean curvature problems through a local minimization principle, Adv. Nonlinear Anal. 2 (2013), no. 4, 427–441. 10.1515/anona-2013-0021Search in Google Scholar

[3] G. A. Afrouzi, S. Heidarkhani and S. Moradi, Existence of weak solutions for three-point boundary-value problems of Kirchhoff-type, Electron. J. Differential Equations (2016) 2016, Paper No. 234. Search in Google Scholar

[4] G. A. Afrouzi, S. Heidarkhani and S. Shokooh, Infinitely many solutions for Steklov problems associated to non-homogeneous differential operators through Orlicz–Sobolev spaces, Complex Var. Elliptic Equ. 60 (2015), no. 11, 1505–1521. 10.1080/17476933.2015.1031122Search in Google Scholar

[5] M. Bohner, G. Caristi, S. Heidarkhani and S. Moradi, A critical point approach to boundary value problems on the real line, Appl. Math. Lett. 76 (2018), 215–220. 10.1016/j.aml.2017.08.017Search in Google Scholar

[6] M. Bohner, G. Caristi, S. Heidarkhani and A. Salari, Three solutions for a class of nonhomogeneous nonlocal systems: An Orlicz–Sobolev space setting, Dynamic Syst. Appl. 26 (2017), 259–282. Search in Google Scholar

[7] G. Bonanno, Relations between the mountain pass theorem and local minima, Adv. Nonlinear Anal. 1 (2012), no. 3, 205–220. 10.1515/anona-2012-0003Search in Google Scholar

[8] G. Bonanno and G. M. Bisci, Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Bound. Value Probl. 2009 (2009), Article ID 670675. 10.1155/2009/670675Search in Google Scholar

[9] G. Bonanno, G. Molica Bisci and V. Rădulescu, Existence of three solutions for a non-homogeneous Neumann problem through Orlicz–Sobolev spaces, Nonlinear Anal. 74 (2011), no. 14, 4785–4795. 10.1016/j.na.2011.04.049Search in Google Scholar

[10] G. Bonanno, G. Molica Bisci and V. Rădulescu, Infinitely many solutions for a class of nonlinear eigenvalue problem in Orlicz–Sobolev spaces, C. R. Math. Acad. Sci. Paris 349 (2011), no. 5–6, 263–268. 10.1016/j.crma.2011.02.009Search in Google Scholar

[11] G. Bonanno, G. Molica Bisci and V. Rădulescu, Arbitrarily small weak solutions for a nonlinear eigenvalue problem in Orlicz–Sobolev spaces, Monatsh. Math. 165 (2012), no. 3–4, 305–318. 10.1007/s00605-010-0280-2Search in Google Scholar

[12] Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), no. 4, 1383–1406. 10.1137/050624522Search in Google Scholar

[13] P. Clément, B. de Pagter, G. Sweers and F. de Thélin, Existence of solutions to a semilinear elliptic system through Orlicz–Sobolev spaces, Mediterr. J. Math. 1 (2004), no. 3, 241–267. 10.1007/s00009-004-0014-6Search in Google Scholar

[14] P. Clément, M. García-Huidobro, R. Manásevich and K. Schmitt, Mountain pass type solutions for quasilinear elliptic equations, Calc. Var. Partial Differential Equations 11 (2000), no. 1, 33–62. 10.1007/s005260050002Search in Google Scholar

[15] G. Dankert, Sobolev embedding theorems in Orlicz spaces, Ph.D. thesis, University of Köln, 1966. Search in Google Scholar

[16] L. Diening, Theoretical and numerical results for electrorheological fluids, Ph.D. thesis, University of Freiburg, 2002. Search in Google Scholar

[17] T. K. Donaldson and N. S. Trudinger, Orlicz–Sobolev spaces and imbedding theorems, J. Funct. Anal. 8 (1971), 52–75. 10.1016/0022-1236(71)90018-8Search in Google Scholar

[18] M. Ferrara and G. Molica Bisci, Existence results for elliptic problems with Hardy potential, Bull. Sci. Math. 138 (2014), no. 7, 846–859. 10.1016/j.bulsci.2014.02.002Search in Google Scholar

[19] M. Galewski and G. Molica Bisci, Existence results for one-dimensional fractional equations, Math. Methods Appl. Sci. 39 (2016), no. 6, 1480–1492. 10.1002/mma.3582Search in Google Scholar

[20] M. García-Huidobro, V. K. Le, R. Manásevich and K. Schmitt, On principal eigenvalues for quasilinear elliptic differential operators: An Orlicz–Sobolev space setting, NoDEA Nonlinear Differential Equations Appl. 6 (1999), no. 2, 207–225. 10.1007/s000300050073Search in Google Scholar

[21] J.-P. Gossez, A strongly nonlinear elliptic problem in Orlicz–Sobolev spaces, Nonlinear Functional Analysis and its Applications (Berkeley 1983), Proc. Sympos. Pure Math. 45 Part 1, American Mathematical Society, Providence (1986), 455–462. 10.1090/pspum/045.1/843579Search in Google Scholar

[22] T. C. Halsey, Electrorheological fluids, Science 258 (1992), no. 5083, 761–766. 10.1126/science.258.5083.761Search in Google Scholar PubMed

[23] S. Heidarkhani, G. A. Afrouzi, M. Ferrara, G. Caristi and S. Moradi, Existence results for impulsive damped vibration systems, Bull. Malays. Math. Sci. Soc. 41 (2018), no. 3, 1409–1428. 10.1007/s40840-016-0400-9Search in Google Scholar

[24] S. Heidarkhani, G. A. Afrouzi, S. Moradi, G. Caristi and B. Ge, Existence of one weak solution for p(x)-biharmonic equations with Navier boundary conditions, Z. Angew. Math. Phys. 67 (2016), no. 3, Article ID 73. 10.1007/s00033-016-0668-5Search in Google Scholar

[25] S. Heidarkhani, G. Caristi and M. Ferrara, Perturbed Kirchhoff-type Neumann problems in Orlicz–Sobolev spaces, Comput. Math. Appl. 71 (2016), no. 10, 2008–2019. 10.1016/j.camwa.2016.03.019Search in Google Scholar

[26] S. Heidarkhani, M. Ferrara, G. A. Afrouzi, G. Caristi and S. Moradi, Existence of solutions for Dirichlet quasilinear systems including a nonlinear function of the derivative, Electron. J. Differential Equations 2016 (2016), Paper No. 56. Search in Google Scholar

[27] S. Heidarkhani, M. Ferrara and G. Caristi, Multiple solutions for perturbed Kirchhoff-type non-homogeneous Neumann problems through Orlicz–Sobolev spaces, Electron. J. Differential Equations 2018 (2018), Paper No. 43. Search in Google Scholar

[28] S. Heidarkhani, Y. Zhou, G. Caristi, G. A. Afrouzi and S. Moradi, Existence results for fractional differential systems through a local minimization principle, Pan-American. Math. J., to appear. 10.1016/j.camwa.2016.04.012Search in Google Scholar

[29] M. A. Krasnosel’skiĭ and J. B. Rutickiĭ, Convex Functions and Orlicz Spaces, P. Noordhoff, Groningen, 1961. Search in Google Scholar

[30] A. Kristály, M. Mihăilescu and V. Rădulescu, Two non-trivial solutions for a non-homogeneous Neumann problem: An Orlicz–Sobolev space setting, Proc. Roy. Soc. Edinburgh Sect. A 139 (2009), no. 2, 367–379. 10.1017/S030821050700025XSearch in Google Scholar

[31] 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), no. 2073, 2625–2641. 10.1098/rspa.2005.1633Search in Google Scholar

[32] M. Mihăilescu and V. Rădulescu, Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: An Orlicz–Sobolev space setting, J. Math. Anal. Appl. 330 (2007), no. 1, 416–432. 10.1016/j.jmaa.2006.07.082Search in Google Scholar

[33] M. Mihăilescu and V. Rădulescu, Neumann problems associated to nonhomogeneous differential operators in Orlicz–Sobolev spaces, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 6, 2087–2111. 10.5802/aif.2407Search in Google Scholar

[34] G. Molica Bisci and V. D. Rădulescu, Bifurcation analysis of a singular elliptic problem modelling the equilibrium of anisotropic continuous media, Topol. Methods Nonlinear Anal. 45 (2015), no. 2, 493–508. 10.12775/TMNA.2015.024Search in Google Scholar

[35] G. Molica Bisci, V. D. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia Math. Appl. 162, Cambridge University, Cambridge, 2016. 10.1017/CBO9781316282397Search in Google Scholar

[36] G. Molica Bisci and R. Servadei, A bifurcation result for non-local fractional equations, Anal. Appl. (Singap.) 13 (2015), no. 4, 371–394. 10.1142/S0219530514500067Search in Google Scholar

[37] G. Molica Bisci and R. Servadei, Lower semicontinuity of functionals of fractional type and applications to nonlocal equations with critical Sobolev exponent, Adv. Differential Equations 20 (2015), no. 7–8, 635–660. 10.57262/ade/1431115711Search in Google Scholar

[38] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer, Berlin, 1983. 10.1007/BFb0072210Search in Google Scholar

[39] B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000), no. 1-2, 401–410. 10.1016/S0377-0427(99)00269-1Search in Google Scholar

[40] M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math. 1748, Springer, Berlin, 2000. 10.1007/BFb0104029Search in Google Scholar

[41] N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–483. 10.1512/iumj.1968.17.17028Search in Google Scholar

[42] V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 4, 675–710, 877; translation in Math. USSR-Izv. 29 (1987), no. 1, 33–66. 10.1070/IM1987v029n01ABEH000958Search in Google Scholar

Received: 2017-01-20
Revised: 2017-12-15
Accepted: 2018-01-15
Published Online: 2019-11-27
Published in Print: 2021-04-01

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