Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the profile of the changing sign mountain pass solutions for an elliptic problem


Authors: E. N. Dancer and Shusen Yan
Journal: Trans. Amer. Math. Soc. 354 (2002), 3573-3600
MSC (2000): Primary 35J60
DOI: https://doi.org/10.1090/S0002-9947-02-03026-X
Published electronically: May 8, 2002
MathSciNet review: 1911512
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider nonlinear elliptic equations with small diffusion and Dirichlet boundary conditions. We construct changing sign solutions with peaks close to the boundary and consider the location of the peak.


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

  • 1. A.Ambrosetti and P.Rabinowitz Dual variational methods in critical point theory and applications, J. Funct. Anal., 14(1973), 349-381. MR 51:6412
  • 2. H.Berestycki, L.Caffarelli and L.Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa C1. Sci., 25, 69-94(1997). MR 2000e:35053
  • 3. D.Cao, E.N.Dancer, E.Noussair and S.Yan, On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems, Discrete and Continuous Dynamical Systems, 2(1996), 221-236.MR 96m:35095
  • 4. D.Cao and T.Küpper, On the existence of the multipeaked solutions to a semilinear Neumman problem, Duke Math. J. 97(1999), 261-300. MR 2000a:35064
  • 5. P.Clément and G.Sweers, Existence and multiplicity results for a semilinear eigenvalue problem, Ann. Scuola Norm. Sup. Pisa, 14(1987), 97-121. MR 89j:35053
  • 6. E.N.Dancer, On the number of positive solutions of weakly non-linear elliptic equations when a parameter is large, Proc. London Math. Soc., 53(1986), 429-452. MR 88c:35061
  • 7. E.N. Dancer, K.Y.Lam and S. Yan, The effect of the graph topology on the existence of multipeak solutions for nonlinear Schrödinger equation, Abstract and Appl. Anal., 3(1998), 293-318. MR 2001g:35231
  • 8. E.N.Dancer and J.Wei, On the profile of solutions with two sharp layers to a singularly perturbed semilinear Dirichlet problem, Proc. Royal Soc. Edinburgh, 127A(1997), 691-701. MR 98i:35012
  • 9. E.N.Dancer and J.Wei, On the location of spikes of solutions with two sharp layers for a singularly perturbed semilinear Dirichlet problem, J. Diff. Equations, 157(1999), 82-101. MR 2000j:35017
  • 10. E.N. Dancer and S. Yan, Multipeak solutions for a singularly perturbed Neumann problem, Pacific J. Math., 189(1999), 241-262. MR 2000d:35010
  • 11. E.N.Dancer and S.Yan, A singularly perturbed elliptic problem in bounded domains with nontrivial topology, Adv. Diff. Equations, 4(1999), 347-368. MR 2000d:35009
  • 12. E.N. Dancer and S. Yan, Interior and boundary peak solutions for a mixed boundary value problem, Indiana University Math. J., 48(1999), 1177-1212. MR 2001f:35146
  • 13. M. Del Pino and P.Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math.J., 48(1999), 883-898. MR 2001b:35027
  • 14. B. Gidas, W.M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68(1979), 209-243.MR 80h:35043
  • 15. D.Gilbarg and N.S.Trudinger, Elliptic partial differential equations of second order, second edition, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983. MR 86c:35035
  • 16. C. Gui and J.Wei, Multiple interior peak solutions for some singularly perturbed Neumann problem, J. Diff. Equations 158(1999), 1-27. MR 2000g:35035
  • 17. C. Gui, J.Wei and M.Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problem, Ann. Inst. H. Poincaré Anal., Non Linéaire, 17(2000), 47-82. MR 2001a:35018
  • 18. E.F.Keller and L.A.Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Bio., 26(1970), 399-415.
  • 19. J.Jang, On the spike solutions of singularly perturbed semilinear Dirichlet problem, J.Diff. Equations, 114(1994), 370-395. MR 95i:35099
  • 20. P.L.Lions, The concentration compactness principle in the calculus of variations, the locally compact case, I,II, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 1(1984), 109-145, 223-283. MR 87e:49035a; MR 87e:49035b
  • 21. Y.Y. Li, On a singularly perturbed equation with Neumann boundary condition, Comm. PDE, 23(1998), 487-545. MR 2000a:35013
  • 22. H.Meinhardt, Models of biological pattern formation, Academic Press, 1982.
  • 23. W.M. Ni and I. Takagi, On the shape of the least energy solution to a semilinear Neumann problem, Comm. Pure Appl. Math. 41(1991), 819-851. MR 92i:35052
  • 24. W.M. Ni and I. Takagi, Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math. J.70(1993), 247-281. MR 94h:35072
  • 25. W.M. Ni, I. Takagi and J.Wei, On the locations and profile of spike-layer solutions to a singularly perturbed semilinear Dirichlet problem, intermediate solution, Duke Math. J., 94(1998), 597-618. MR 99h:35011
  • 26. W.M.Ni and J.Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math. 48(1995), 731-768. MR 96g:35077
  • 27. G.Sweers, Some results for a semilinear elliptic problem with a large parameter, Proceedings of the First International Conference on Industrial and Applied Mathematics, Paris-La Villette, 1987, pp. 109-116. MR 89d:35065
  • 28. J.Wei, On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem, J. Diff. Equations, 129(1996), 315-333. MR 97f:35015
  • 29. S. Yan, On the number of the interior multipeak solutions for singularly perturbed Neumann problems, Topological Methods in Nonlinear Anal., 12(1999), 61-78. MR 2001c:35024

Similar Articles

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

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


Additional Information

E. N. Dancer
Affiliation: School of Mathematics and Statistics, University of Sydney, New South Wales 2006, Australia
Email: normd@maths.usyd.edu.au

Shusen Yan
Affiliation: School of Mathematics and Statistics, University of Sydney, New South Wales 2006, Australia
Email: shusen@maths.usyd.edu.au

DOI: https://doi.org/10.1090/S0002-9947-02-03026-X
Received by editor(s): December 4, 2001
Received by editor(s) in revised form: February 19, 2002
Published electronically: May 8, 2002
Additional Notes: This work was supported by the ARC
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society