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)

 
 

 

Some remarks related to De Giorgi's conjecture


Authors: Yihong Du and Li Ma
Journal: Proc. Amer. Math. Soc. 131 (2003), 2415-2422
MSC (2000): Primary 35J15, 35J60
DOI: https://doi.org/10.1090/S0002-9939-02-06867-3
Published electronically: November 27, 2002
MathSciNet review: 1974639
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For several classes of functions including the special case $f(u)=u-u^3$, we obtain boundedness and symmetry results for solutions of the problem $-\Delta u=f(u)$ defined on $R^n$. Our results complement a number of recent results related to a conjecture of De Giorgi.


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

  • [AAC] G. Alberti, L. Ambrosio and X. Cabre, On a long-standing conjecture of E. De Giorgi: old and recent results, to appear in Acta Applicandae Mathematicae.
  • [AC] L. Ambrosio and X. Cabre, Entire solutions of semi-linear elliptic equations in $R^3$ and a conjecture of De Giorgi, Journal of Amer. Math. Soc., 13(2000), 725-739. MR 2001g:35064
  • [AW] D.G. Aronson and H.F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30(1978), 33-76. MR 80a:35013
  • [BBG] M.T. Barlow, R.F. Bass and C. Gui, The Liouville property and a conjecture of De Giorgi, Comm. Pure Appl. Math., 53(2000), 1007-1038. MR 2001m:35095
  • [BCN1] H. Berestycki, L. Caffarelli and L. Nirenberg, Monotonicity for elliptic equations in an unbounded Lipschitz domain, Comm. Pure Appl. Math., 50(1997), 1089-1111. MR 98k:35064
  • [BCN2] H. Berestycki, L. Caffarelli and L. Nirenberg, Further properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)25(1997), 69-94. MR 2000e:35053
  • [BHM] H. Berestycki, F. Hamel, and R. Monneau, One-dimensional symmetry of bounded entire solutions of some elliptic equations, Duke Math. J, 103(2000), 375-396. MR 2001j:35069
  • [CGS] L. Caffarelli, N. Garofalo, and F. Segala, A gradient bound for entire solutions of quasi-linear equations and its consequences, Comm. Pure Appl. Math., 47(1994), 1457-1473. MR 95k:35030
  • [DD] E.N. Dancer and Y. Du, Some remarks on Liouville type results for quasilinear elliptic equations, to appear in Proc. Amer. Math. Soc.
  • [dG] E. De Giorgi, Convergence problems for functionals and operators, Proc. Int. Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), E. de Giorgi et al. (eds.), Pitagora, Bologna (1979), pp. 131-188. MR 80k:49010
  • [DM] Y. Du and L. Ma, Logistic type equations on $R^N$ by a squeezing method involving boundary blow-up solutions, J. London Math. Soc., 64(2001), 107-124. MR 2002d:35089
  • [F1] A. Farina, Symmetry for solutions of semi-linear elliptic equations in $R^N$ and related conjectures, Rend. Mat. Acc. Lincei, 10(1999), 255-265. MR 2001g:35072
  • [F2] A. Farina, Finite-energy solutions, quantization effects and Liouville-type results for a variant of the Ginzburg-Landau systems in $R^k$, Diff. Integral Eqns., 11(1998), 975-893. MR 99j:35199
  • [Fr] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, New Jersey, 1964. MR 31:6062
  • [GG] N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann., 311(1998), 481-491. MR 99j:35049
  • [GT] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin/New York, 1983. MR 86c:35035; reprint MR 2001k:35004
  • [Ke] J.B. Keller, On solutions of $\Delta u=f(u)$, Comm. Pure Appl. Math., 10(1957), 503-510. MR 19:964c
  • [M] L. Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations, Comm. Pure Appl. Math., 38(1985), 679-684. MR 87m:35088
  • [MM] L. Modica and S. Mortola, Some entire solutions in the plane of nonlinear Poisson equations, Boll. Un. Mat. Ital., B, 17(1980), 614-622. MR 81k:35036

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35J15, 35J60

Retrieve articles in all journals with MSC (2000): 35J15, 35J60


Additional Information

Yihong Du
Affiliation: School of Mathematical and Computer Sciences, University of New England, Armidale, New South Wales 2351, Australia
Email: ydu@turing.une.edu.au

Li Ma
Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
Email: lma@math.tsinghua.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-02-06867-3
Keywords: Elliptic equation, maximum principle, symmetry of solution
Received by editor(s): March 10, 2002
Published electronically: November 27, 2002
Additional Notes: The first author was partially supported by the Australian Academy of Science and Academia Sinica under an exchange program while part of this work was carried out. The second author was partially supported by a grant from the national 973 project of China and a scientific grant of Tsinghua University at Beijing.
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society