Home Existence results for a Kirchhoff type equation in Orlicz–Sobolev spaces
Article
Licensed
Unlicensed Requires Authentication

Existence results for a Kirchhoff type equation in Orlicz–Sobolev spaces

  • EL Miloud Hssini EMAIL logo , Najib Tsouli and Mustapha Haddaoui
Published/Copyright: March 16, 2017

Abstract

In this paper, based on the mountain pass theorem and Ekeland’s variational principle, we show the existence of solutions for a class of non-homogeneous and nonlocal problems in Orlicz–Sobolev spaces.

MSC 2010: 34B27; 35J60; 35B05

References

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

[2] G. A. Afrouzi, V. Rădulescu and S. Shokooh, Multiple solutions of Neumann problems: An Orlicz–Sobolev space setting, Bull. Malays. Math. Sci. Soc. (2015), 10.1007/s40840-015-0153-x. 10.1007/s40840-015-0153-xSearch in Google Scholar

[3] C. O. Alves, F. J. S. A. Corrêa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005), 85–93. 10.1016/j.camwa.2005.01.008Search in Google Scholar

[4] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381. 10.1016/0022-1236(73)90051-7Search in Google Scholar

[5] G. Autuori, P. Pucci and M. C. Salvatori, Asymptotic stability for anisotropic Kirchhoff systems, J. Math. Anal. Appl. 352 (2009), 149–165. 10.1016/j.jmaa.2008.04.066Search in Google Scholar

[6] G. Bonanno and G. Molica Bisci, Existence of three solutions for a non-homogeneous Neumann problem through Orlicz–Sobolev spaces, Nonlinear Anal. 74 (2011), 4785–4795. 10.1016/j.na.2011.04.049Search in Google Scholar

[7] 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), 263–268. 10.1016/j.crma.2011.02.009Search in Google Scholar

[8] 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

[9] F. Cammaroto and L. Vilasi, Multiple solutions for a non-homogeneous Dirichlet problem in Orlicz–Sobolev spaces, Appl. Math. Comput. 218 (2012), 11518–11527. Search in Google Scholar

[10] N. T. Chung, Thee solutions for a class of nonlocal problems in Orlicz–Sobolev space, J. Korean Math. Soc. 50 (2013), no. 6, 1257–1269. 10.4134/JKMS.2013.50.6.1257Search in Google Scholar

[11] N. T. Chung, Existence of solutions for nonlocal problems in Orlicz–Sobolev spaces via genus theory, Acta Univ. Apulensis Math. Inform. 37 (2014), 111–123. Search in Google Scholar

[12] N. T. Chung and H. Q. Toan, On a nonlinear and non-homogeneous problem without (A-R) type condition in Orlicz–Sobolev spaces, Appl. Math. Comput. 219 (2013), 7820–7829. 10.1016/j.amc.2013.02.011Search 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), 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), 33–62. 10.1007/s005260050002Search in Google Scholar

[15] F. Colasuonno and P. Pucci, Multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal. 74 (2011), 5962–5974. 10.1016/j.na.2011.05.073Search in Google Scholar

[16] F. J. S. A. Corrêa and G. M. Figueiredo, On a p-Kirchhoff equation via Krasnoselskii’s genus, Appl. Math. Lett. 22 (2009), 819–822. 10.1016/j.aml.2008.06.042Search in Google Scholar

[17] G. Dai, Three solutions for a nonlocal Dirichlet boundary value problem involving the p(x)-Laplacian, Appl. Anal. 92 (2013), no. 1, 191–210. 10.1080/00036811.2011.602633Search in Google Scholar

[18] G. Dai and R. Ma, Solutions for a p(x)-Kirchhoff type equation with Neumann boundary data, Nonlinear Anal. 12 (2011), 2666–2680. 10.1016/j.nonrwa.2011.03.013Search in Google Scholar

[19] M. Dreher, The Kirchhoff equation for the p-Laplacian, Rend. Semin. Mat. Univ. Politec. Torino 64 (2006), 217–238. Search in Google Scholar

[20] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353. 10.1016/0022-247X(74)90025-0Search in Google Scholar

[21] X. Fan and D. Zhao, On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl. 263 (2001), 424–446. 10.1006/jmaa.2000.7617Search in Google Scholar

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

[23] G. M. Figueiredo and J. A. Santos, On a nonlocal multivalued problem in an Orlic–Sobolev space via Krasnoselskii’s genus, preprint (2015), https://arxiv.org/abs/1501.05596v1. Search in Google Scholar

[24] M. Garciá-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), 207–225. 10.1007/s000300050073Search in Google Scholar

[25] J. P. Gossez and R. Manàsevich, On a nonlinear eigenvalue problem in Orlicz–Sobolev spaces, Proc. Roy. Soc. Edinburgh Sect. A 132 (2002), 891–909. 10.1017/S030821050000192XSearch in Google Scholar

[26] N. Halidias, An Orlicz–Sobolev space setting for quasilinear elliptic problems, Electron. J. Differential Equations 2005 (2005), no. 29, 1–7. Search in Google Scholar

[27] X. He and W. Zou, Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal. 70 (2009), no. 3, 1407–1414. 10.1016/j.na.2008.02.021Search in Google Scholar

[28] E. M. Hssini, M. Massar, M. Talbi and N. Tsouli, Infinitely many solutions for nonlocal elliptic p-Kirchhoff type equation under Neumann boundary condition, Int. J. Math. Anal. 7 (2013), no. 21, 1011–1022. 10.12988/ijma.2013.13101Search in Google Scholar

[29] E. M. Hssini, M. Massar and N. Tsouli, Existence and multiplicity of solutions for a p(x)-Kirchhoff type problems, Bol. Soc. Parana. Mat. (3) 33 (2015), no. 2, 201–215. Search in Google Scholar

[30] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. Search in Google Scholar

[31] 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), 367–379. 10.1017/S030821050700025XSearch in Google Scholar

[32] V. K. Le, A global bifurcation result for quasilinear elliptic equations in Orlicz–Sobolev spaces, Topol. Methods Nonlinear Anal. 15 (2000), no. 2, 301–327. 10.12775/TMNA.2000.022Search in Google Scholar

[33] M. Massar, M. Talbi and N. Tsouli, Multiple solutions for nonlocal system of (p(x),q(x))-Kirchhoff type, Appl. Math. Comput. 242 (2014), 216–226. Search in Google Scholar

[34] M. Mihăilescu and V. Rădulescu, Neumann problems associated to nonhomogeneous differential operators in Orlicz–Sobolev space, Ann. Inst. Fourier (Grenoble) 6 (2008), 2087–2111. 10.5802/aif.2407Search in Google Scholar

[35] M. Mihăilescu and V. Rădulescu, A continuous spectrum for nonhomogeneous differential operators in Orlicz–Sobolev spaces, Math. Scand. 104 (2009), 132–146. 10.7146/math.scand.a-15090Search in Google Scholar

[36] M. Mihăilescu and D. Repovš, Multiple solutions for a nonlinear and non-homogeneous problem in Orlicz–Sobolev spaces, Appl. Math. Comput. 217 (2011), 6624–6632. 10.1016/j.amc.2011.01.050Search in Google Scholar

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

[38] M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer, Berlin, 1996. 10.1007/978-3-662-03212-1Search in Google Scholar

[39] L. Yang, Multiplicity of solutions for perturbed nonhomogeneous Neumann problem through Orlicz–Sobolev spaces, Abstr. Appl. Anal. 2012 (2012), Article ID 236712. 10.1155/2012/236712Search in Google Scholar

[40] Y. Yang and J. Zhang, A note on the existence of solutions for a class of quasilinear elliptic equations: An Orlicz–Sobolev space setting, Bound. Value Probl. 2012 (2012), Paper No. 136. 10.1186/1687-2770-2012-136Search in Google Scholar

[41] E. Zeidler, Nonlinear Functional Analysis and its Applications. Vol. II/B, Berlin, New York, 1985. 10.1007/978-1-4612-5020-3Search in Google Scholar

Received: 2016-7-4
Revised: 2017-1-23
Accepted: 2017-1-27
Published Online: 2017-3-16
Published in Print: 2017-7-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 29.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/apam-2016-0065/html
Scroll to top button