DOI QR코드

DOI QR Code

THREE SOLUTIONS FOR A CLASS OF NONLOCAL PROBLEMS IN ORLICZ-SOBOLEV SPACES

  • Nguyen, Thanh Chung (Department of Science Management & International Cooperation Quang Binh University)
  • 투고 : 2012.11.11
  • 발행 : 2013.11.01

초록

Using the three critical points theorem by B. Ricceri [23], we obtain a multiplicity result for a class of nonlocal problems in Orlicz-Sobolev spaces. To our knowledge, this is the first contribution to the study of nonlocal problems in this class of functional spaces.

키워드

참고문헌

  1. R. Adams, Sobolev Spaces, Academic Press, New York, 1975.
  2. A. Bensedik and M. Bouchekif, On an elliptic equation of Kirchhoff-type with a potential asymptotically linear at infinity, Math. Comput. Modelling 49 (2009), no, 5-6, 1089-1096. https://doi.org/10.1016/j.mcm.2008.07.032
  3. F. Cammaroto and L. Vilasi, Multiple solutions for a Kirchhoff-type problem involving the p(x)-Laplacian operator, Nonlinear Anal. 74 (2011), no. 5, 1841-1852. https://doi.org/10.1016/j.na.2010.10.057
  4. F. Cammaroto and L. Vilasi, Multiple solutions for a nonhomogeneous Dirichlet problem in Orlicz-Sobolev spaces, Appl. Math. Comput. 218 (2012), no. 23, 11518-11527. https://doi.org/10.1016/j.amc.2012.05.039
  5. C. Y. Chen, Y. C. Kuo, and T. F.Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations 250 (2011), no. 4, 1876-1908. https://doi.org/10.1016/j.jde.2010.11.017
  6. M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal. 30 (1997), no. 7, 4619-4627. https://doi.org/10.1016/S0362-546X(97)00169-7
  7. N. T. Chung, Multiple solutions for a p(x)-Kirchhoff-type equation with sign-changing nonlinearities, Complex Variables and Elliptic Equations, 2012, 1-10, iFirst.
  8. N. T. Chung, Multiplicity results for a class of p(x)-Kirchhoff type equations with combined nonlinearities, Electron. J. Qual. Theory Differ. Equ. 2012 (2012), no. 42, 1-13. https://doi.org/10.1186/1687-1847-2012-1
  9. Ph. Clement, M. Garcia-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
  10. Ph. Clement, B. dePagter, G. Sweers, and F. deTh?in, Existence of solutions to a semilinear elliptic system through Orlicz-Sobolev spaces, Mediterr. J. Math. 1 (2004), no. 3, 241-267. https://doi.org/10.1007/s00009-004-0014-6
  11. F. Colasuonno and P. Pucci, Multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal. 74 (2011), no. 17, 5962-5974. https://doi.org/10.1016/j.na.2011.05.073
  12. F. J. S. A. Correa and G. M. Figueiredo, On a p-Kirchhoff equation via Krasnoselskii's genus, Appl. Math. Letters 22 (2009), no. 6, 819-822. https://doi.org/10.1016/j.aml.2008.06.042
  13. G. Dai and R. Hao, Existence of solutions for a p(x)-Kirchhoff-type equation, J. Math. Anal. Appl. 359 (2009), no. 1, 275-284. https://doi.org/10.1016/j.jmaa.2009.05.031
  14. X. L. 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
  15. F. Fang and Z. Tan, Existence and Multiplicity of solutions for a class of quasilinear elliptic equations: An Orlicz-Sobolev setting, J. Math. Anal. Appl. 389 (2012), no. 1, 420-428. https://doi.org/10.1016/j.jmaa.2011.11.078
  16. G. Kirchhoff, Mechanik, Teubner, Leipzig, Germany, 1883.
  17. J. Lamperti, On the isometries of certain function-spaces, Pacific J. Math. 8 (1958), 459-466. https://doi.org/10.2140/pjm.1958.8.459
  18. D. Liu, On a p-Kirchhoff equation via fountain theorem and dual fountain theorem, Nonlinear Anal. 72 (2010), no. 1, 302-308. https://doi.org/10.1016/j.na.2009.06.052
  19. T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal. 63 (2005), 1967-1977. https://doi.org/10.1016/j.na.2005.03.021
  20. M. Mihailescu and D. Repovs, Multiple solutions for a nonlinear and non-homogeneous problem in Orlicz-Sobolev spaces, Appl. Math. Comput. 217 (2011), 6624-6632. https://doi.org/10.1016/j.amc.2011.01.050
  21. M. Mihailescu and V. Radulescu, Neumann problems associated to non-homogeneous differential operators in Orlicz-Sobolev spaces, Ann. Inst. Fourier 6 (2008), no. 6, 2087- 2111.
  22. M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker Inc., New York, 1991.
  23. B. Ricceri, A further three critical points theorem, Nonlinear Anal. 71 (2009), no. 9, 4151-4157. https://doi.org/10.1016/j.na.2009.02.074
  24. B. Ricceri, On an elliptic Kirchhoff-type problem depending on two parameters, J. Global Optim. 46 (2010), no. 4, 543-549. https://doi.org/10.1007/s10898-009-9438-7
  25. J. J. Sun and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type equations, Nonlinear Anal. 74 (2011), no. 4, 1212-1222. https://doi.org/10.1016/j.na.2010.09.061
  26. E. Zeidler, Nonlinear Functional Analysis and Applications, In Nonlinear monotone operators, Vol. II/B, Springer-Verlag, New York, 1990.

피인용 문헌

  1. Multiple solutions for Kirchhoff elliptic equations in Orlicz-Sobolev spaces vol.2017, pp.1, 2017, https://doi.org/10.1186/s13661-017-0865-y
  2. Existence of least energy nodal solution with two nodal domains for a generalized Kirchhoff problem in an Orlicz-Sobolev space vol.290, pp.4, 2017, https://doi.org/10.1002/mana.201500286