Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the equation $ {\rm div}\,(\vert \nabla u\vert \sp {p-2}\nabla u)+\lambda\vert u\vert \sp {p-2}u=0$


Author: Peter Lindqvist
Journal: Proc. Amer. Math. Soc. 109 (1990), 157-164
MSC: Primary 35J60; Secondary 35P05
DOI: https://doi.org/10.1090/S0002-9939-1990-1007505-7
Addendum: Proc. Amer. Math. Soc. 116 (1992), 583-584.
MathSciNet review: 1007505
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The first eigenvalue $ \lambda = {\lambda _1}$ for the equation $ \operatorname{div} ({\text{\vert}}\nabla u{{\text{\vert}}^{p - 2}}\nabla u{\text{) + }}\lambda {\text{\vert}}u{{\text{\vert}}^{p - 2}}u = 0$ is simple in any bounded domain. (Through the nonlinear counterpart to the Rayleigh quotient $ {\lambda _1}$ is related to the Poincaré inequality.)


References [Enhancements On Off] (What's this?)

  • [1] A. Anane, Simplicité et isolation de la première valeur propre du $ p$-laplacien avec poids, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), 725-728. MR 920052 (89e:35124)
  • [2] J. García Azorero and I. Peral Alonso, Existence and non-uniqueness for the $ p$-Laplacian: Non-linear eigenvalues, Comm. Partial Differential Equations 12 (1987), 1389-1430. MR 912211 (89e:35058)
  • [3] E. DiBenedetto, $ {C^{1 + \alpha }}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), 827-850. MR 709038 (85d:35037)
  • [4] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Springer-Verlag, Berlin-Heidelberg, 1983. MR 737190 (86c:35035)
  • [5] G. Mostow, Quasi-conformal mappings in $ n$-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 53-104. MR 0236383 (38:4679)
  • [6] G. Pólya and G. Szegö, Isoperimetric inequalities in mathematical physics, Princeton Univ. Press, Princeton, NJ, 1951. MR 0043486 (13:270d)
  • [7] F. Riesz and B. Sz.-Nagy, Vorlesungen über Funktionalanalysis, Deutscher Verlag der Wissenschaften, Berlin, 1956. MR 0083695 (18:747e)
  • [8] S. Sakaguchi, Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems, Ann. Scuola. Norm. Sup. Pisa Cl. Sci. (4) (to appear). MR 951227 (89h:35133)
  • [9] S. Sobolev, Applications of functional analysis in mathematical physics, American Mathematical Society, Providence, RI, 1963. MR 0165337 (29:2624)
  • [10] F. de Thélin, Quelques résultais d'existence et de non-existence pour une E. D. P. elliptique non linéaire, C. R. Acad. Sci. Paris. Sér. I Math. 299 (1984), 911-914.
  • [11] -, Sur l'espace propre associé à la première valeur propre du pseudo-laplacien, C. R. Acad. Sci. Paris Sér. I. Math. 303 (1986), 355-358. MR 860838 (87i:35147)
  • [12] P. Tolksdorf, Regularity for a more general class of quasi-linear elliptic equations, J. Differential Equations 51 (1984), 126-150. MR 727034 (85g:35047)
  • [13] N. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721-747. MR 0226198 (37:1788)
  • [14] L. Veron and M. Guedda, Quasilinear elliptic equations involving critical Sobolev exponents, Université François Rabelais (Tours), 1988. (to appear in Nonlinear Anal.) MR 1009077 (90h:35100)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 35J60, 35P05

Retrieve articles in all journals with MSC: 35J60, 35P05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-1007505-7
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society