Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

DIRICHLET EIGENVALUE PROBLEMS UNDER MUSIELAK-ORLICZ GROWTH

  • Received : 2021.11.05
  • Accepted : 2022.08.30
  • Published : 2022.11.01
Download PDF
⟨ Previous Next ⟩

Abstract

This paper studies the eigenvalues of the G(·)-Laplacian Dirichlet problem $$\{-div\;\(\frac{g(x,\;{\mid}{\nabla}u{\mid})}{{\mid}{\nabla}u{\mid}}{\nabla}u\)={\lambda}\;\(\frac{g(x,{\mid}u{\mid})}{{\mid}u{\mid}}u\)\;in\;{\Omega}, \\u\;=\;0\;on\;{\partial}{\Omega},$$ where Ω is a bounded domain in ℝN and g is the density of a generalized Φ-function G(·). Using the Lusternik-Schnirelmann principle, we show the existence of a nondecreasing sequence of nonnegative eigenvalues.

Keywords

References

  1. H. Amann, Lusternik-Schnirelman theory and non-linear eigenvalue problems, Math. Ann. 199 (1972), 55-72. https://doi.org/10.1007/BF01419576
  2. L. Barbu, G. Morosanu, and C. Pintea, A nonlinear elliptic eigenvalue-transmission problem with Neumann boundary condition, Ann. Mat. Pura Appl. (4) 198 (2019), no. 3, 821-836. https://doi.org/10.1007/s10231-018-0801-5
  3. A. Benyaiche and I. Khlifi, Sobolev-Dirichlet problem for quasilinear elliptic equations in generalized Orlicz-Sobolev spaces, Positivity 25 (2021), no. 3, 819-841. https://doi.org/10.1007/s11117-020-00789-z
  4. F. E. Browder, Existence theorems for nonlinear partial differential equations, in Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968), 1-60, Amer. Math. Soc., Providence, RI, 1970.
  5. R. Chiappinelli, What do you mean by nonlinear eigenvalue problems?, Axioms. 7(2018), 39. https://doi.org/10.3390/axioms7020039
  6. X. Fan, Q. Zhang, and D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem, J. Math. Anal. Appl. 302 (2005), no. 2, 306-317. https://doi.org/10.1016/j.jmaa.2003.11.020
  7. J. P. Garc'ia Azorero and I. Peral Alonso, Existence and nonuniqueness for the pLaplacian: nonlinear eigenvalues, Comm. Partial Differential Equations 12 (1987), no. 12, 1389-1430. https://doi.org/10.1080/03605308708820534
  8. P. Harjulehto and P. Hasto, Orlicz spaces and generalized Orlicz spaces, Lecture Notes in Mathematics, 2236, Springer, Cham, 2019. https://doi.org/10.1007/978-3-030-15100-3
  9. A. Le, Eigenvalue problems for the p-Laplacian, Nonlinear Anal. 64 (2006), no. 5, 1057-1099. https://doi.org/10.1016/j.na.2005.05.056
  10. G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations, Comm. Partial Differential Equations 16 (1991), no. 2-3, 311-361. https://doi.org/10.1080/03605309108820761
  11. J. Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics, 1034, Springer-Verlag, Berlin, 1983. https://doi.org/10.1007/BFb0072210
  12. V. D. Radulescu and S. Saiedinezhad, A nonlinear eigenvalue problem with p(x)-growth and generalized Robin boundary value condition, Commun. Pure Appl. Anal. 17 (2018), no. 1, 39-52. https://doi.org/10.3934/cpaa.2018003
  13. M. Tienari, Lusternik-Schnirelmann theorem for the generalized Laplacian, J. Differential Equations 161 (2000), no. 1, 174-190. https://doi.org/10.1006/jdeq.2000.3712
  14. E. Zeidler, The Lusternik-Schnirelmann theory for indefinite and not necessarily odd nonlinear operators and its applications, Nonlinear Anal. 4 (1980), no. 3, 451-489. https://doi.org/10.1016/0362-546X(80)90085-1
  15. E. Zeidler, Nonlinear Functional Analysis and Its Applications. III, translated from the German by Leo F. Boron, Springer-Verlag, New York, 1985. https://doi.org/10.1007/978-1-4612-5020-3