Previous |  Up |  Next

Article

Keywords:
non-homogeneous Neumann problem; variational methods; Orlicz--Sobolev space
Summary:
Under a suitable oscillatory behavior either at infinity or at zero of the nonlinear term, the existence of infinitely many weak solutions for a non-homogeneous Neumann problem, in an appropriate Orlicz--Sobolev setting, is proved. The technical approach is based on variational methods.
References:
[1] Adams R. A.: Sobolev Spaces. Pure and Applied Mathematics, 65, Academic Press, New York, 1975. MR 0450957 | Zbl 1098.46001
[2] Afrouzi G. A., Graef J. R., Shokooh S.: Multiple solutions for Neumann systems in an Orlicz–Sobolev space setting. Miskolc Math. Notes 18 (2017), no. 1, 31–45. DOI 10.18514/MMN.2017.1906 | MR 3669881
[3] Afrouzi G. A., Heidarkhani S., Shokooh S.: 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. MR 3393865
[4] Afrouzi G. A., Rădulescu V., Shokooh S.: Multiple solutions of Neumann problems: an Orlicz–Sobolev space setting. Bull. Malays. Math. Sci. Soc. 40 (2017), no. 4, 1591–1611. DOI 10.1007/s40840-015-0153-x | MR 3712573
[5] Bonanno G.: A critical point theorem via the Ekeland variational principle. Nonlinear Anal. 75 (2012), no. 5, 2992–3007. DOI 10.1016/j.na.2011.12.003 | MR 2878492
[6] Bonanno G., Bisci G. M., Rădulescu V.: Existence of three solutions for a non-homogeneous Neumann problem through Orlicz–Sobolev spaces. Nonlinear Anal. 74 (2011), no. 14, 4785–4795. DOI 10.1016/j.na.2011.04.049 | MR 2810717
[7] Bonanno G., Bisci G. M., Rădulescu V.: 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. DOI 10.1016/j.crma.2011.02.009 | MR 2783317
[8] Bonanno G., Bisci G. M., Rădulescu V.: Arbitrarily small weak solutions for a nonlinear eigenvalue problem in Orlicz–Sobolev spaces. Monatsh. Math. 165 (2012), no. 3–4, 305–318. DOI 10.1007/s00605-010-0280-2 | MR 2891255
[9] Bonanno G., Bisci G. M., Rădulescu V.: Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz–Sobolev spaces. Nonlinear Anal. 75 (2012), no. 12, 4441–4456. DOI 10.1016/j.na.2011.12.016 | MR 2927113
[10] Bonanno G., Candito P.: Infinitely many solutions for a class of discrete non-linear boundary value problems. Appl. Anal. 88 (2009), no. 4, 605–616. DOI 10.1080/00036810902942242 | MR 2541143
[11] Bonanno G., Di Bella B.: Infinitely many solutions for a fourth-order elastic beam equation. NoDEA Nonlinear Differential Equations Appl. 18 (2011), no. 3, 357–368. DOI 10.1007/s00030-011-0099-0 | MR 2811057
[12] Chabrowski J., Fu Y.: Existence of solutions for $p(x)$-Laplacian problems on a bounded domain. J. Math. Anal. Appl. 306 (2005), no. 2, 604–618. DOI 10.1016/j.jmaa.2004.10.028 | MR 2136336
[13] Clément Ph., de Pagter B., Sweers G., de Thélin F.: Existence of solutions to a semilinear elliptic system through Orlicz–Sobolev spaces. Mediterr. J. Math. 1 (2004), no. 3, 241–267. DOI 10.1007/s00009-004-0014-6 | MR 2094464
[14] D'Aguì G., Sciammetta A.: Infinitely many solutions to elliptic problems with variable exponent and nonhomogeneous Neumann conditions. Nonlinear Anal. 75 (2012), no. 14, 5612–5619. DOI 10.1016/j.na.2012.05.009 | MR 2942940
[15] Diening L.: Maximal function on Musielak–Orlicz spaces and generalized Lebesgue spaces. Bull. Sci. Math. 129 (2005), no. 8, 657–700. MR 2166733
[16] Fan X.: Solutions for $p(x)$-Laplacian Dirichlet problems with singular coefficients. J. Math. Anal. Appl. 312 (2005), no. 2, 464–477. DOI 10.1016/j.jmaa.2005.03.057 | MR 2179089
[17] Fan X., Zhao D.: On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$. J. Math. Anal. Appl. 263 (2001), no. 2, 424–446. MR 1866056
[18] Fan X., Zhang Q., Zhao D.: Eigenvalues of $p(x)$-Laplacian Dirichlet problem. J. Math. Anal. Appl. 302 (2005), no. 2, 306–317. DOI 10.1016/j.jmaa.2003.11.020 | MR 2107835
[19] Halsey T. C.: Electrorheological fluids. Science 258 (1992), 761–766. DOI 10.1126/science.258.5083.761
[20] Kováčik O., Rákosník J.: On spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$. Czechoslovak Math. J. 41 (1991), no. 4, 592–618. MR 1134951
[21] Kristály A., Mihăilescu M., Rădulescu V.: 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. MR 2496969
[22] Mihăilescu M., Rădulescu V.: Eigenvalue problems associated with nonhomogeneous differential operators in Orlicz–Sobolev spaces. Anal. Appl. (Singap.) 6 (2008), no. 1, 83–98. DOI 10.1142/S0219530508001067 | MR 2380887
[23] Mihăilescu M., Rădulescu V.: Neumann problems associated to nonhomogeneous differential operators in Orlicz–Sobolev space. Ann. Inst. Fourier (Grenoble) 58 (2008), no. 6, 2087–2111. DOI 10.5802/aif.2407 | MR 2473630
[24] Musielak J.: Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics, 1034, Springer, Berlin, 1983. MR 0724434 | Zbl 0557.46020
[25] Pfeiffer C., Mavroidis C., Cohen Y. B., Dolgin B.: Electrorheological fluid based force feedback device. Conference on Telemanipulator and Telepresence Technologies VI, Part of SPIE's Photonics East, Boston, Proc. 3840 (1999), 88–99.
[26] Rao M. M., Ren Z. D.: Theory of Orlicz Spaces. Monographs and Textbooks in Pure and Applied Mathematics, 146, Marcel Dekker, New York. 1991. MR 1113700 | Zbl 0724.46032
[27] Ricceri B.: A general variational principle and some of its applications. J. Comput. Appl. Math. 113 (2000), no. 1–2, 401–410. DOI 10.1016/S0377-0427(99)00269-1 | MR 1735837
[28] Růžička M.: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics, 1748, Springer, Berlin, 2000. DOI 10.1007/BFb0104030 | MR 1810360 | Zbl 0968.76531
Partner of
EuDML logo