Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON A NONLOCAL PROBLEM WITH INDEFINITE WEIGHTS IN ORLICZ-SOBOLEV SPACE

  • Received : 2019.02.04
  • Accepted : 2019.10.14
  • Published : 2020.04.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we consider a class of nonlocal problems with indefinite weights in Orlicz-Sobolev space. Under some suitable conditions on the nonlinearities, we establish some existence results using variational techniques and Ekeland's variational principle.

Keywords

References

  1. E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal. 164 (2002), no. 3, 213-259. https://doi.org/10.1007/s00205-002-0208-7
  2. R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
  3. K. Ait-Mahiout and C. O. Alves, Existence and multiplicity of solutions for a class of quasilinear problems in Orlicz-Sobolev spaces without Ambrosetti-Rabinowitz condition, J. Elliptic Parabol. Equ. 4 (2018), no. 2, 389-416. https://doi.org/10.1007/s41808-018-0026-1
  4. B. Amaziane, L. Pankratov, and A. Piatnitski, Nonlinear flow through double porosity media in variable exponent Sobolev spaces, Nonlinear Anal. Real World Appl. 10 (2009), no. 4, 2521-2530. https://doi.org/10.1016/j.nonrwa.2008.05.008
  5. M. Avci, Ni-Serrin type equations arising from capillarity phenomena with non-standard growth, Bound. Value Probl. 2013 (2013), 55, 13 pp. https://doi.org/10.1186/1687-2770-2013-55
  6. M. Avci, Existence results for anisotropic discrete boundary value problems, Electron. J. Differential Equations 2016 (2016), Paper No. 148, 11 pp.
  7. M. Avci, On a nonlocal Neumann problem in Orlicz-Sobolev spaces, J. Nonlinear Funct. Anal. 2017 (2017), Article ID 42, 1-11. https://doi.org/10.23952/jnfa.2017.42
  8. M. Avci and A. Pankov, Nontrivial solutions of discrete nonlinear equations with variable exponent, J. Math. Anal. Appl. 431 (2015), no. 1, 22-33. https://doi.org/10.1016/j.jmaa.2015.05.056
  9. M. Avci and B. Suer, Existence results for some nonlocal problems involving variable exponent, J. Elliptic Parabolic Equations 2019 (2019). https://doi.org/10.1007/s41808-018-0032-3
  10. R. Ayazoglu, M. Avci, and N. T. Chung, Existence of solutions for nonlocal problems in Orlicz-Sobolev spaces via monotone method, Electron. J. Math. Anal. Appl. 4 (2016), no. 1, 63-73.
  11. P. Blomgren, T. F Chan, P. Mulet, and C. K. Wong, Total variation image restoration: numerical methods and extensions, in Proceedings of the International Conference on Image Processing, IEEE, 3 1997, 384-387.
  12. M.-M. Boureanu and D. N. Udrea, Existence and multiplicity results for elliptic problems with p(${\cdot}$)-growth conditions, Nonlinear Anal. Real World Appl. 14 (2013), no. 4, 1829-1844. https://doi.org/10.1016/j.nonrwa.2012.12.001
  13. B. Cekic, A. V. Kalinin, R. A. Mashiyev, and M. Avci, $L^{p(x)}({\Omega})$-estimates of vector fields and some applications to magnetostatics problems, J. Math. Anal. Appl. 389 (2012), no. 2, 838-851. https://doi.org/10.1016/j.jmaa.2011.12.029
  14. 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. https://doi.org/10.1137/050624522
  15. N. T. Chung, Three solutions for a class of nonlocal problems in Orlicz-Sobolev spaces, J. Korean Math. Soc. 50 (2013), no. 6, 1257-1269. https://doi.org/10.4134/jkms.2013.50.6.1257
  16. N. T. Chung, Multiple solutions for a nonlocal problem in Orlicz-Sobolev spaces, Ric. Mat. 63 (2014), no. 1, 169-182. https://doi.org/10.1007/s11587-013-0171-7
  17. N. T. Chung, Existence of solutions for a class of Kirchhoff type problems in Orlicz-Sobolev spaces, Ann. Polon. Math. 113 (2015), no. 3, 283-294. https://doi.org/10.4064/ap113-3-5
  18. Ph. Clement, M. G. Huidobro, R. Manasevich, and K. Schmitt, Mountain pass type solutions for quasilinear elliptic equations, Calc. Var. Partial Differential Equations 11 (2000), no. 1, 33-62. https://doi.org/10.1007/s005260050002
  19. F. J. S. A. Correa and G. M. Figueiredo, On a p-Kirchhoff equation via Krasnoselskii's genus, Appl. Math. Lett. 22 (2009), no. 6, 819-822. https://doi.org/10.1016/j.aml.2008.06.042
  20. D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces, Applied and Numerical Harmonic Analysis, Birkhauser/Springer, Heidelberg, 2013. https://doi.org/10.1007/978-3-0348-0548-3
  21. L. Diening, P. Harjulehto, P. Hasto, and M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, 2017, Springer, Heidelberg, 2011. https://doi.org/10.1007/978-3-642-18363-8
  22. I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324-353. https://doi.org/10.1016/0022-247X(74)90025-0
  23. X. Fan, On nonlocal p(x)-Laplacian Dirichlet problems, Nonlinear Anal. 72 (2010), no. 7-8, 3314-3323. https://doi.org/10.1016/j.na.2009.12.012
  24. F. Fang and Z. Tan, Existence and multiplicity of solutions for a class of quasilinear elliptic equations: an Orlicz-Sobolev space setting, J. Math. Anal. Appl. 389 (2012), no. 1, 420-428. https://doi.org/10.1016/j.jmaa.2011.11.078
  25. M. Garcia-Huidobro, V. K. Le, R. Manasevich, 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. https://doi.org/10.1007/s000300050073
  26. B. Ge, On an eigenvalue problem with variable exponents and sign-changing potential, Electron. J. Qual. Theory Differ. Equ. 2015 (2015), Paper No. 92, 10 pp. https://doi.org/10.14232/ejqtde.2015.1.92
  27. 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. https://doi.org/10.1016/j.camwa.2016.03.019
  28. H. Hudzik, On generalized Orlicz-Sobolev space, Funct. Approximatio Comment. Math. 4 (1976), 37-51.
  29. O. Kovacik and J. Rakosnik, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J. 41(116) (1991), no. 4, 592-618.
  30. M. Mihailescu and V. Radulescu, Neumann problems associated to nonhomogeneous differential operators in Orlicz-Sobolev spaces, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 6, 2087-2111. https://doi.org/10.5802/aif.2407
  31. M. Mihailescu and V. Radulescu, Eigenvalue problems associated with nonhomogeneous differential operators in Orlicz-Sobolev spaces, Anal. Appl. (Singap.) 6 (2008), no. 1, 83-98. https://doi.org/10.1142/S0219530508001067
  32. M. Mihailescu, V. Radulescu, and D. Repovs, On a non-homogeneous eigenvalue problem involving a potential: an Orlicz-Sobolev space setting, J. Math. Pures Appl. (9) 93 (2010), no. 2, 132-148. https://doi.org/10.1016/j.matpur.2009.06.004
  33. J. Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics, 1034, Springer-Verlag, Berlin, 1983. https://doi.org/10.1007/BFb0072210
  34. M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, 146, Marcel Dekker, Inc., New York, 1991.
  35. M. Ruzicka, Electrorheological fluids: modeling and mathematical theory, Lecture Notes in Mathematics, 1748, Springer-Verlag, Berlin, 2000. https://doi.org/10.1007/BFb0104029
  36. A. K. Souayah, On a class of nonhomogenous quasilinear problems in Orlicz-Sobolev spaces, Opuscula Math. 32 (2012), no. 4, 731-750. https://doi.org/10.7494/OpMath.2012.32.4.731
  37. Z. Yucedag, M. Avci, and R. Mashiyev, On an elliptic system of p(x)-Kirchhoff-type under Neumann boundary condition, Math. Model. Anal. 17 (2012), no. 2, 161-170. https://doi.org/10.3846/13926292.2012.655788
  38. V. V. Zhikov, Differential Equations, Differ. Equ. 33 (1997), no. 1, 108-115; translated from Differ. Uravn. 33 (1997), no. 1, 107-114, 143.