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THE FIRST EIGENVALUE OF SOME (p, q)-LAPLACIAN AND GEOMETRIC ESTIMATES

  • Azami, Shahroud (Department of Mathematics Faculty of Sciences Imam Khomeini International University)
  • Received : 2017.04.04
  • Accepted : 2017.08.17
  • Published : 2018.01.31

Abstract

We study the nonlinear eigenvalue problem for some of the (p, q)-Laplacian on compact manifolds with zero boundary condition. In particular, we obtain some geometric estimates for the first eigenvalue.

References

  1. A. Abolarinwa, The first eigenvalue of p-Laplacian and gometric estimates, Nonl. Anal-ysis and Differential Equations 2 (2014), no. 3, 105-115. https://doi.org/10.12988/nade.2014.455
  2. N. Benouhiba and Z. Belyacine, A class of eigenvalue problems for the (p; q)-Laplacian in RN, Int. J. Pure Appl. Math. 80 (2012), no. 5, 727-737.
  3. I. Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, 115, Academic Press, Inc., Orlando, FL, 1984.
  4. S. Y. Cheng, Eigenfunctions and eigenvalues of Laplacian, in Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 2, 185-193, Amer. Math. Soc., Providence, RI.
  5. Q.-M. Cheng and H. Yang, Estimates on eigenvalues of Laplacian, Math. Ann. 331 (2005), no. 2, 445-460. https://doi.org/10.1007/s00208-004-0589-z
  6. E. M. Harrell, II and P. L. Michel, Commutator bounds for eigenvalues, with applications to spectral geometry, Comm. Partial Differential Equations 19 (1994), no. 11-12, 2037-2055. https://doi.org/10.1080/03605309408821081
  7. S. Kawai and N. Nakauchi, The first eigenvalue of the p-Laplacian on a compact Rie-mannian manifold, Nonlinear Anal. 55 (2003), no. 1-2, 33-46. https://doi.org/10.1016/S0362-546X(03)00209-8
  8. A. El Khalil, S. El Manouni, and M. Ouanan, Simplicity and stability of the first eigen-value of a nonlinear elliptic system, Int. J. Math. Math. Sci. 2005, no. 10, 1555-1563.
  9. P. F. Leung, On the consecutive eigenvalues of the Laplacian of a compact minimal submanifold in a sphere, J. Austral. Math. Soc. Ser. A 50 (1991), no. 3, 409-416. https://doi.org/10.1017/S1446788700033000
  10. P. Li, Geometric analysis, Cambridge Studies in Advanced Mathematics, 134, Cam-bridge University Press, Cambridge, 2012.
  11. A.-M. Matei, First eigenvalue for the p-Laplace operator, Nonlinear Anal. 39 (2000), no. 8, Ser. A: Theory Methods, 1051-1068. https://doi.org/10.1016/S0362-546X(98)00266-1
  12. H. Takeuchi, On the first eigenvalue of the p-Laplacian in a Riemannian manifold, Tokyo J. Math. 21 (1998), no. 1, 135-140. https://doi.org/10.3836/tjm/1270041991