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)

 

 

An alternate variational principle for $ \Delta u-u+\mid u\mid \sp{r-1}{\rm sgn}\,u=0$


Author: Charles V. Coffman
Journal: Proc. Amer. Math. Soc. 87 (1983), 666-670
MSC: Primary 35J60; Secondary 35J20
MathSciNet review: 687637
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: An alternate variational principle for the equation in the title has been proposed by H. A. Levine. We analyse the relation between this principle and the Rayleigh quotient that has been used previously for the variational study of this problem in $ {R^N}$. The main result is an existence theorem for $ {W^{1,2}}({R^N})$-solutions of the variational problem posed by Levine.


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

  • [1] Charles V. Coffman, Uniqueness of the ground state solution for Δ𝑢-𝑢+𝑢³=0 and a variational characterization of other solutions, Arch. Rational Mech. Anal. 46 (1972), 81–95. MR 0333489
  • [2] Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR 0445088
  • [3] H. A. Levine, A numerical estimate for the best constant in a two-dimensional Sobolev inequality with three integral norms, Notices Amer. Math. Soc. 26 (1979), Abstract 79T-B16.
  • [4] Howard A. Levine, An estimate for the best constant in a Sobolev inequality involving three integral norms, Ann. Mat. Pura Appl. (4) 124 (1980), 181–197. MR 591555, 10.1007/BF01795392
  • [5] Zeev Nehari, On a nonlinear differential equation arising in nuclear physics, Proc. Roy. Irish Acad. Sect. A 62 (1963), 117–135 (1963). MR 0165176
  • [6] Giorgio Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353–372. MR 0463908

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 35J60, 35J20

Retrieve articles in all journals with MSC: 35J60, 35J20


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0687637-X
Keywords: Variational principle, nonlinear elliptic boundary value problem
Article copyright: © Copyright 1983 American Mathematical Society