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Quasilinearity below the 1st eigenvalue


Author: Victor L. Shapiro
Journal: Proc. Amer. Math. Soc. 129 (2001), 1955-1962
MSC (2000): Primary 35J60; Secondary 35J35
DOI: https://doi.org/10.1090/S0002-9939-01-06124-X
Published electronically: February 15, 2001
MathSciNet review: 1825902
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Abstract:

This paper establishes the existence of two nontrivial weak solutions for a quasilinear Dirichlet problem below the first eigenvalue via the mountain pass theorem.


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

  • [Ad] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. MR 56:9247
  • [Br] F. E. Browder, Existence theorems for nonlinear partial differential equations, Proc.Sympos. Pure Math. Vol. 16 , Amer. Math. Soc., Providence, R. I., 1970, pp. 1-60. MR 42:4855
  • [KS] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York (1980). MR 81g:49013
  • [GT] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Springer-Verlag, Berlin, 1983. MR 86c:35035
  • [LL] J. Leray and J.-L. Lions, Quelques résultats de Visik sur les problèmes elliptiques nonlinéaires par les méthodes Minty-Browder, Bull. Soc. Math. France 93 (1965), 97-107. MR 33:2939
  • [Ra] P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conf. Ser. in Math. , No. 65, Amer. Math. Soc., Providence, R. I., 1986. MR 87j:58024
  • [RS] A. Rumbos and V. L. Shapiro, One-sided resonance for a quasilinear variational problem, Contemporary Math. Vol. 208 , Amer. Math. Soc., Providence, R. I., 1997, pp. 277-299. MR 98j:35074
  • [Sh] V. L. Shapiro, Quasilinear ellipticity and the first eigenvalue, Comm. Partial Differential Equations 16 (1991), 1819-1855. MR 92j:35053

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Additional Information

Victor L. Shapiro
Affiliation: Department of Mathematics, University of California, Riverside, California 92521-0135
Email: shapiro@math.ucr.edu

DOI: https://doi.org/10.1090/S0002-9939-01-06124-X
Keywords: Quasilinear variational, first eigenvalue, mountain pass theorem, near $p$-homogeneous.
Received by editor(s): September 7, 1999
Published electronically: February 15, 2001
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2001 American Mathematical Society

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